~_n_n—  n_-r  —  n  —  n, 


REESE  LIBRARY 

OF  THE 

UNIVERSITY  OF  CALIFORNIA.' 


Deceived  A/y  ,-89 

ssion  No.  /  Q  6  3  0    .   C/js.v  No. 


THE    ELEMENTS 


OF 


WATER    SUPPLY 


ENGINEERING. 


BY 


E,    SHERMAN   GOULD, 


M.    AM.    SOC.    C.    E. 


NEW  YORK: 

THE  ENGINEERING  NEWS  PUBLISHING  CO. 
1899. 


COPYRIGHTED,  1899, 
BY  THE  ENGINEERING  NEWS  PUBLISHING  CO, 

72%  30 


PREFACE. 


"  Practical  Hydraulic  Formula? "  was  first  published  in 
1880.  The  second  edition  appeared  in  189±  in  a  form  much 
extended  by  notes  on  THE  QUALITY  and  THE  QUANTITY  OF  WATER 
and  on  THE  CALCULATION"  and  THE  CONSTRUCTION  OF  DAMS. 

In  the  present  work  all  the  matter  contained  in  the  first  and 
second  editions  of  "Practical  Hydraulic  Formulas"  is  repub- 
lished  ;  but,  under  the  heading  of  "  NOTES  TO  PARTS  I.  AND  II.," 
it  has  been  supplemented  with  copious  memoranda,  elucidating 
and  greatly  extending  many  important  points  inadequately 
treated  in  the  previous  editions. 

An  entirely  new  part  (PART  II I.)  has  been  added.    This  treats 

Of   the   FLOW    OF    WATER    THROUGH    MASONRY    CONDUITS   Oil   the 

basis  of  Darcy's  formulae,  with  some  practical  details  of  aqueduct 
and  tunnel  construction.  A  few  paragraphs  are  devoted  to  the 
subject  of  the  FILTRATION"  OF  PUBLIC  WATER  SUPPLIES,  suffi- 
cient, it  is  hoped,  to  indicate  the  proper  lines  of  further  investi- 
gation to  those  who  are  interested  in  pursuing  them.  A  some- 
what fuller  treatment  is  given  to  the  subject  of  PUMPING  EN- 
GINKS  AND  DUTY  TRIALS.  Some  pages,  have  been  added  on  the 
subject  of  ARCHES  AND  ABUTMENTS.  As  an  extended  project  of 
water  supply  frequently  embraces  the  construction  of  arched 
masonry  aqueducts,  it  is  believed  that  the  brief  and  practical 
rules  given  in  this  part  of  the  book  will  prove  acceptable  to  the 
hydraulic  engineer.  A  set  of  carefully  calculated,  labor-saving 
TABLES,  with  explanations  and  examples,  terminates  the  book. 

It  will  be  seen  that  the  present  work  covers  so  wide  a  field 
that  to  retain  for  it  as  a  whole  the  title  originally  given  to  the 
first  part  would  be  misleading.  It  is,  therefore,  called  "  THE 
ELEMENTS  OF  WATER  SUPPLY  ENGINEERING,"  which  name 
more  truly  indicates  its  scope. 

While  it  is  believed  that  every  topic  connected  with  water- 
supply  engineering  has  been  at  least  touched  upon,  especial  pains 
have  been  taken  to  go  into  very  close  detail  in  the  matter  of  the 
principal  dimensions  and  quantities  involved  in  the  designing  of 
hydraulic  work.  As  these  are  the  points  which  the  author  has 
most  carefully  sought  for  in  his  own  reading  and  observation,  so 
,  he  believes  that  they  are  the  ones  which  others  may  be  most  in- 
terested in  finding  fully  treated  of  in  the  present  volume. 

E.  SHERMAN  GOULD. 


OF  THE 

UNIVERSITY 


TABLE  OF  CONTENTS, 

INTRODUCTION. 
CHAPTER  I. 

Flow  through  a  short  horizontal  pipe— Effect  on  velocity  of  increased 
length— Frictional  head— Hydraulic  grade  line— Hydrostratic  and  hy- 
draulic pressures— Piezometric  tubes— Results  of  raising  a  pipe  line  above 
the  hydraulic  grade  line— Why  the  water  ceases  to  rise  in  the  upper 
stories  of  the  houses  of  a  town  when  the  consumption  is  increased— In- 
fluence of  inside  surface  of  pipes  upon  velocity  of  flow— Darcy's  coef- 
ficients— Fundamental  equations — Length  of  a  pipe  line  usually  deter- 
mined by  its  horizontal  projection— Numerical  examples  of  simple  and 
compound  system^ Pages  11-24 

CHAPTER  II. 

Calculations  are  the  same  for  pipes  laid  horizontally  or  on  a  slope— 
Qualification  of  this  statement — Pipe  of  uniform  diameter  equivalent 
to  compound  system— General  formula— Numerical  example— Use  of 
logarithms  (foot  note)— Numerical  example  of  branch  pipe — Simplified 
method— Numerical  examples— Relative  discharges  through  branches 
variously  placed— Discharges  determined  by  plotting — Caution  regarding 
results  obtained  by  calculation — Numerical  examples  •  Pages  25-38 

CHAPTER  III. 

Numerical  example  of  a  system  of  pipes  for  the  supply  of  a  town— Es- 
tablishment of  additional  formulae  for  facilitating  such  calculations — 
Determination  of  diameters— Pumping  and  reservoirs — Caution  regard- 
ing calculated  results— Useful  approximate  formulae —Table  of  5th 
powers — Preponderating  influence  of  diameter  over  grade  illustrated  by 
example  Pages  39-48 

CHAPTER  IV. 

Use  of  formula  14  illustrated  by  numerical  example  of  compound 
system  combined  with  branches — Comparison  of  results — Rough  and 


8  TABLE   OF  CONTENTS. 

smooth  pipes— Pipes  communicating  with  three  reservoirs— Numerical 
examples  under  varying  conditions— Loss  of  head  from  other  causes 
than  friction— Velocity,  entrance  and  exit  head— Numerical  examples 
and  general  formula— Downward  discharge  through  a  vertical  pipe- 
Other  minor  losses  of  head— Abrupt  changes  of  diameter— Partially 
opened  valve — Branches  and  bends — Centrifugal  force — Small  import- 
ance of  all  losses  of  head  except  frictional  in  the  case  of  long  pipes — 
All  such  covered  by  "  even  inches"  in  the  diameter  -  Pages  49-63 

CHAPTER  V. 

Notes  on  pipelaying       -  Pages  64-67 

APPENDIX. 

Weight  of  cast  iron  pipe — Various  useful  formulae        •        Pages  68-70 


SECOND  PART. 

• 
NOTES  ON  WATER  SUPPLY  ENGINEERING. 

Quality  of  water— Quantity  of  .water— Dams,   calculation  of;   con- 
struction of— Reference  to  other  .publications         -       -         Pages  71- 105 

Notes  to  Parts  I  and  II       -       .....       .       .       Pages  106-122 


THIRD  PART. 

Flow  of  Water  Through  Masonry  Conduits      —       -  Pages  123-125 

Some  Details  of  Tunnel  and  Acqueduct  Construction  Pages  125-126 

Filtration  of  Public  Water  Supplies       -  Pages  127-129 

Pumping  Engines  and  Duty  Trials         •       -       -       -  Pages  129-137 

Arches  and  Abutments       -------  Pages  137-154 

Hydraulic  Tables          *    .   -       * Pages  154-162 

Index       -       - Pages  163-168 


INTRODUCTION  TO  HYDRAULIC  FORMULA. 


The  following  pages  first  appeared  as  a  series  of  articles  in 
the  columns  of  ENGINEERING  NEWS.  They  are  now  repub- 
lished  with  a  few  corrections  and  additions. 

In  virtue  of  the  law  of  gravitation,  water  tends  naturally  to 
pass  from  a  higher  to  a  lower  level,  and  without  a  difference  of 
level  there  can  be  no  natural  flow. 

It  can  be  said  in  all  seriousness — although  the  statement 
may  seem  to  invite  the  unjust  accusation  of  an  ill-timed  attempt 
at  pleasantry — that  the  whole  science  of  hydraulics  is  founded 
upon  the  three  following  homely  and  unassailable  axioms  : 

First.  That  water  always  seeks  its  own  lowest  level. 

Second.  That,  therefore,  it  always  tends  to  run  down  hill, 
and 

Third.  That,  other  things  being  equal,  the  steeper  the  hill, 
the  faster  it  runs. 

In  the  case  of  water  flowing  through  long  pipes,  the  hill 
down  which  it  tends  to  run  is  the  HYDRAULIC  GRADE  LINE.  If 
the  pipe  be  of  uniform  diameter  and  character,  the  hydraulic 
grade  line  is  a  straight  line  joining  the  water  surfaces  at  its  two 
extremities,  provided  that  the  pipe  lies  wholly  below  such 
straight  line,  and  its  declivity  is  measured — like  that  of  all  hills 
— by  the  ratio  of  its  height  to  its  length. 

But  if  there  be  any  changes  whatever  in  the  pipe,  either  in 
diameter  or  in  the  nature  of  its  inside  surface  ;  or  if  there  be  in- 

(9) 


10  INTRODUCTION    TO    HYDRAULIC   FOUMILJE. 

crease  or  diminution  of  the  volume  of  water  entering  it  at  its 
upper  extremity  by  reason  of  branches  leading  to  or  from  the 
main  pipe,  then  the  hydraulic  grade  line  becomes  broken  and 
distorted  to  a  greater  or  less  extent,  so  that  its  declivity  is  not 
uniform  from  end  to  end,  but  consists  of  a  series  of  varying 
grades  some  steeper  than  others  though  all  sloping  in  the  same 
direction. 

As  regards  the  third  axiom,  the  proviso — "  other  things  being 
equal" — must  not  be  overlooked.  For  we  shall  find  that  a  pipe 
of  greater  diameter  but  less  hydraulic  declivity  than  another, 
may  give  a  greater  velocity  to  the  water  passing  through  it. 
Also,  of  two  pipes  of  the  same  hydraulic  slope  and  diameter,  the 
one  having  the  smoother  inside  surface  affords  the  greater 
velocity. 

The  vertical  distance  from  any  point  in  a  pipe  to  the  hy- 
draulic grade  line,  constitutes  the  Piezometric  JieigJtt,  and  meas- 
ures the  hydraulic  pressure  at  that  point.  It  will  be  seen  that 
the  solution  of  problems  relating  to  the  flow  of  water  through 
pipes,  lies  in  the  knowing  or  ascertaining  of  the  piezometric 
height  at  any  desired  point.  In  general,  it  is  necessary  to  es- 
tablish the  piezometric  height  for  every  point  of  change  of  any 
kind  which  occurs  throughout  the  entire  length  of  the  conduit. 
The  joining  of  the  upper  extremities  of  these  heights  gives  the 
complete  hydraulic  grade  line. 

The  object  of  the  following  papers  is  to  establish  systematic 
methods  for  tracing  the  hydraulic  grade  line  under  the  different 
circumstances  likely  to  occur  in  practice,  and  generally,  to  fur- 
nish solutions  for  a  large  number  of  practical  problems,  com- 
mencing with  the  simplest  cases  and  extending  to  some  rather 
intricate  ones,  not  usually  embraced  in  our  hydraulic  manuals. 

E.  S.  G. 
SCRANTON,  Pa.,  May,  1889. 


HYDRAULIC  FORMULAE. 


CHAPTER  I. 

Flow  through  a  Short  Horizontal  Pipe—  Effect  on  Velocity  of  Incr,  ased  Length— 
Frictional  Head—  Hydraulic  Grade  Line  -  Hydrostatic  and  Hydraulic  Pres- 
sures— Piezometric  Tubes—  Result  of  liaising  a  I  'ipe  Line  Above  the  Hydraulic 
Grade  Line—  Why  the  Water  Ceases  to  Rise  in  the  Upper  Stories  of  the  Houses 
of  a  Town  when  the  Consumption  is  Increased—  Influence  of  Inside  Surface  of 
Pipes  Upon  Velocity  of  Flow—  Darcy's  Coefficients—  Fundamental  Equations- 
Length  of  a  Pipe  Line  Usually  Determined  by  its  Horizontal  Projection—  Nu- 
merical Examples  of  Simple  and  Compound  Systems. 

Let  us  suppose  a  reservoir  of  large  relative  area  and  depth  to 
be  tapped  near  its  bottom  by  a  horizontal  cylindrical  pipe,  of 
which  the  length  is  equal  to  about  three  times  its  diameter. 

If  there  were  no  physical  resistance  to  the  flow,  the  velocity 
of  the  water  issuing  from  the  pipe  would  be  given  by  the  formula 
for  the  velocity  of  falling  bodies  . 


in  which  V  =  velocity  in  feet  per  second,  g  =  the  acceleration 
due  to  gravity  =  32.2  ft.,  and  H  =  the  height,  expressed  in  feet, 
of  the  surface  of  the  water  in  the  reservoir  above  the  center  of  the 

Pipe- 

Observation  shows,  however,  that  in  the  case  cited  the  ve- 
locity of  discharge  is  equal  only  to  that  theoretically  due  to  a 
height  of  about  two-thirds  of  H:,  that  is  : 


!?_£?  =  6.55  yjj. 


12  PRACTICAL  HYDRAULIC  FORMULA. 

The  remaining  third  of  the  height  is  consumed  in  over- 
coming the  resistance  offered  to  entry  by  the  edges  of  the  orifice 
to  the  inflowing  vein  of  water.  The  head  necessary  to  overcome 
the  resistance  to  entry  is  therefore  about  one-half  of  that  neces- 
sary to  produce  the  velocity  of  flow. 

If  the  length  of  the  pipe  should  be  increased  progressively 
and  indefinitely,  the  velocity  would  be  found  to  diminish  in- 
versely as  the  square  root  of  the  length.  It  would  correspond, 
therefore,  to  a  smaller  and  smaller  percentage  of  the  total  head 
H.  The  resistance  to  entry  diminishes  directly  as  the  velocity, 
and  the.  head  necessary  to  overcome  it  is  always  equal  to  about 
one-half  of  that  necessary  to  produce  the  given  velocity  as  cal- 
culated by  the  laws  of  falling  bodies. 

As  the  length  of  the  pipe  (always  supposed  to  remain  hori- 
zontal) increases,  and  the  velocity  of  discharge  diminishes,  the 
sum  of  these  two  heads,  i.  e.,  one  and  a  half  times  that  necessary 
to  produce  the  actual  velocity,  is  no  longer  equal  to  the  total 
head  H,  as  we  have  seen  to  be  the  case  when  the  length  of  the 
pipe  is  only  about  three  diameters.  What,  then,  becomes  of  the 
remainder  of  #?  It  is  consumed  in  overcoming  the  increasing 
frictional  resistances  engendered  by  contact  of  the  Coving  water 
with  the  inside  surface  of  the  pipe.  When  the  pipe  is  very  long, 
and  the  velocity  therefore  relatively  low,  the  sum  of  the  velocity 
and  entrance  heads  is  small,  and  by  far  the  greater  part  of  the 
total  head  is  required  to  force  the  water  through  the  pipe  against 
the  opposition  offered  by  friction  to  its  flow.  In  such  cases, 
which  are  those  occurring  most  generally  in  practice  when  water 
is  conveyed  from  a  reservoir  for  the  supply  of  a  town,  the  velocity 
and  entrance  heads  are  commonly  ignored,  and  the  total  head  If 
is  supposed  to  be  available  for  overcoming  the  frictional  resist- 
ances. As  this  occasions,  however,  an  error — although  generally 
a  very  small  one — in  the  wrong  direction,  judgment  is  required 
in  exercising  this  latitude.  Later  on  we  will  revert  to  this  point, 


PRACTICAL    HYDRAULIC    FORMULAE.  13 

but  for  the  present  we  will  consider  only  frictional  resistances, 
particularly  since — and  indeed  because — in  practice  our  assumed 
data  are  almost  always  sufficient  to  afford  an  ample  margin  to 
cover  the  neglected  factors. 

In  what  precedes  we  have  considered  a  horizontal  pipe  issuing 
from  a  reservoir  in  which  the  surface  of  the  water  is  main- 
tained at  a  constant  level.  In  practice  these  conditions  rarely 
obtain. 


FIG.  i. 

Suppose  a  system,  such  as  is  shown  by  Fig.  1,  consisting  of 
a  reservoir  and  pipe  line  of  varying  and  contrary  slopes.  As  the 
level  of  the  water  in  the  reservoir  would  be  subject  to  fluctua- 
tions, and  liable  at  times  to  be  greatly  drawn  down,  it  is  custom- 
ary to  consider  the  surface  of  the  water  as  standing  at  its  lowest 
possible  level,  i.  e.,  the  mouth  of  the  pipe.  In  this  case,  the 
value  of  H  would  be  equal  to  the  difference  of  level  of  the  two 
extremities  a  and  b  of  the  pipe,  and  the  line  ab  joining  the 
centers  of  the  two  ends  would  form  what  is  called  thv  hydraulic 
grade  line,  the  establishing  of  which  is  the  first  step  to  be  taken 
in  laying  out  a  system  of  water  supply. 

Suppose  that  at  the  points  c,  d,  and  e  vertical  tubes,  open  at 
their  upper  ends,  were  connected  with  the  pipe.  The  water, 
when  flowing  freely  from  the  end  b  of  the  pipe  would  rise  in  each 
of  these  tubes  to  about  the  height  of  the  hydraulic  grade  line  at 
these  points,  and,  if  branches  were  connected  at  the  points  c,  d, 
and  e,  they  would,  when  closed,  sustain  a  pressure  upon  their 


14  PRACTICAL  HYDRAULIC   FORMULAE. 

gates  equal  to  the  bead  comprised  between  the  gates  and  the  grade 
line.  If  the  gates  were  open,  the  branches  would  discharge  water 
under  heads  equal  to  the  difference  of  level  of  the  hydraulic  grade 
line  at  the  point  of  embranchment  and  their  remote  extremities, 
less  a  certain  amount  depending  upon  the  volume  discharged, 
which  will  be  spoken  of  hereafter. 

At  d,  where  the  top  of  the  pipe  just  touches  the  grade  line, 
there  would  be  no  pressure  at  all  when  the  water  was  flowing 
through  the  pipe,  except  the  very  small  amount  due  to  the  depth 
of  water  in  the  pipe  itself. 

If  the  end  b  should  be  closed  so  that  there  was  no  movement 
of  water  in  the  pipe,  the  water  would  rise  in  the  tubes,  if  they 
were  long  enough,  until  it  stood  at  the  same  level  as  the  water  in 
the  reservoir,  and  the  pressures  at  c,  d,  and  e  would  be  equal  to 
the  head  comprised  between  these  points  and  the  level  of  the 
water  in  the  reservoir.  The  latter  is  called  the  hydrostatic  press- 
ure, or  simply  the  static  pressure,  and  the  former  the  hydraulic 
pressure,  at  these  points. 

The  tubes  spoken  of  are  known  by  the  name  of  piezometric 
tithes.  . 

The  importance  of  correctly  establishing  the  hydraulic  grade 
fine  is  illustrated  by  reference  to  a  case  such  as  is  shown  in  Fig.  2, 
in  which  the  pipe,  at  the  point  c,  rises  above  the  grade  line  ab. 
To  explain:  It  will  be  readily  deduced  from  what  has  been  al- 
ready said  in  reference  to  horizontal  pipes  that  the  velocity  of  flow, 
and. consequently  the  delivery,  of  a  pipe  increases  with  the  steep- 
ness of  its  slope.  In  this  case  the  pipe  ab  is  divided  into  two  parts, 
the  one  ac  with  a  hydraulic  grade  line  flatter  than  ab,  and  the 
other  cl  with  one  steeper  than  ab.  The  delivery  of  the  entire 
system,  if  the  pipe  were  of  the  same  diameter  throughout,  would 
be  governed  by  the  flatter  portion  ac,  and  the  portion  cb  would 
be  capable,  in  virtue  of  its  steeper  slope,  of  discharging  a  greater 


PRACTICAL  HYDRAULIC   FORMULA.  15 

volume  of  water  than  it  could  receive  from  ac.  Consequently  it 
would  act  merely  as  a  trough  and  would  never  run  full,  and  if  a 
piezometric  tube  were  placed  in  it  at  d  for  instance,  no  water 
would  rise  in  the  tube,  and  no  pressure  be  exerted. 


FIG.  2. 


It  is  very  important,  therefore,  in  locating  a  pipe  line  that 
the  pipe  should  nowhere  rise  above  the  hydraulic  grade  line.  The 
full  amount  of  water  could  indeed  be  carried  over  the  high  point 
c  by  means  of  siphonage,  but  this  expedient  is  not  resorted  to  in 
practice.  Should  the  nature  of  the  ground  require  such  a  location 
as  that  shown  in  Fig.  2,  it  would  be  necessary  to  increase  the  diam- 
eter of  the  pipe  between  a  and  c,  so  that  it  would  deliver  the  re- 
quired volume  under  the  reduced  head,  and  to  diminish  that  be- 
tween c  and  b,  so  that  it  should  only  deliver  the  same  volume 
under  its  increased  head,  and  therefore  run  full.  The  calculations 
necessary  to  determine  the  proper  diameters  will  be  shortly  de- 
veloped. 

Should  the  axis  of  the  pipe  coincide  exactly  with  the  hy- 
draulic grade  line  ab,  the  pipe  would  run  full  (provided  the  feed 
were  sufficient)  but  would  be  under  no  pressure,  and  no  water 
would  rise  in  piezometric  tubes  placed  on  any  part  of  the  pipe. 
Moreover,  as  the  slope  would  be  the  same  for  any  portion  of  the 
pipe,  the  velocity  and  delivery  would  be  unchanged,  whether  we 


16  PRACTICAL  HYDRAULIC   FORMULAE. 

cut  the  pipe  off  at  a  comparatively  short  length,  or  extend  it  in- 
definitely. 

As  a  further  and  very  interesting  practical  illustration  of  the 
effects  of  a  hydraulic  grade  line  of  varying  steepness,  let  us  con- 


FIG.  3. 


sider  (Fig.  3)  the  case  of  a  house  supplied  with  water  by  a  pipe 
communicating  with  a  reservoir. 

Suppose  the  pipe  to  be  sufficiently  large  to  furnish  a  certain 
volume  of  water  per  hour  to  the  upper  story  of  the  house.  If 
now  a  larger  volume  were  required,  it  is  clear  that,  unless  we  in- 
crease the  diameter  of  the  pipe,  it  would  be  necessary  to  increase 
the  steepness  of  pitch  of  the  grade  line,  in  other  words,  to  in- 
crease the  head,  or  difference  of  level  between  the  reservoir  and 
the  point  of  discharge.  The  increased  volume  could  therefore 
be  only  drawn  from  a  lower  story.* 

Or,  to  put  the  same  conditions  under  a  different  form,  sup- 
pose, as  before,  the  pipe  to  be  just  large  enough  to  supply  the 
top  story  of  the  house,  the  taps  on  the  lower  floors  being  closed. 
Should  they  be  opened,  it  is  evident  that  a  greater  amount  of  water 
would  be  discharged  from  them  than  from  the  upper  one,  because 
they  would  discharge  under  a  greater  head.  The  result  would 
be  a  diminished  flow  or  perhaps  no  flow  at  all  on  the  top  floor, 
and  an  increased  discharge  of  water  at  a  lower  level. 

*  In  other  words,  if  we  wish  to  increase  fhe  volume,  the  diameter  of  pipe 
rem «» in inj?  constant,  we  must  increase  the  velocity;  and  the  increased  veJocit^  can 
only  be  obtained  by  an  increased  difference  ot'  level  between  the  two  ends  of  the 
pipe.  If  the  elevation  of  the  upper  end,  or  surface  of  water  in  the  reservoir,  cannot 
be  increased,  that  of  the  1  jwer  end,  or  point  of  discharge,  must  be  diminished. 


PRACTICAL   HYDRAULIC   FORMULA.  ..17 

This  case  shows  why  the  water  ceases  to  rise  in  the  up- 
per stories  of  the  houses  of  a  town  when  the  consumption  in- 
creases. 

It  has  been  found  by  observation  that  the  velocity  of  water 
flowing  through  pipes  is  greatly  affected  by  the  nature  of  their 
inside  surface,  increasing  with  the  smoothness  and  diminishing 
with  the  roughness  of  the  same.  By  direct  experiment,  coeffi- 
cients have  been  established  for  different  conditions  of  surface. 
It  has  also  been  found  that  these  coefficients  vary  slightly  with  the 
diameter  of  the  pipe,  a  pipe  of  a  certain  size  giving  a  greater  ve- 
locity than  one  of  the  same  character  of  inside  surface  but  of 
smaller  diameter,  the  differences  becoming  smaller  as  the  diame- 
ters increase. 

The  value  of  this  coefficient,  which  will  be  designated 
throughout  this  paper  by  C,  is  given  below  for  a  number  of  dif- 
ferent diameters  and  for  two  classes  of  pipes, — those  which  are 
clean  and  smooth  on  the  inside,  and  those  which  are  rough  and 
incrusted,  the  difference  being  as  2  to  1.  As  all  pipes,  after  a  few 
years  of  service,  are  liable  to  become  more  or  less  roughened  and  ob- 
structed by  deposits,  it  is  always  safer  when  calculating  the  proper 
diameters  of  a  permanent  water  supply,  to  assume  rough  pipes 
at  once,  although  diameters  thus  calculated  will,  for  perhaps  a 
number  of  years,  deliver  quantities  greatly  in  excess  of  the  de- 
sired amounts. 

The  coefficients  given  below  are  those  determined  experi- 
mentally by  DARCY.  Of  course,  in  the  subsequent  calculations 
which  will  be  made,  any  other  values  might  be  substituted  for 
the  ones  given.  It  is  well  to  remark,  however,  in  regard  to  the 
coefficient,  that  although  this  factor  is  a  controlling  one  in  the 
calculation  of  the  discharge  of  pipes,  it  is  useless  to  attempt  an 
excessive  refinement  in  establishing  its  value,  because  not  only  is 
it  difficult  to  determine  this  value  with  exactness  for  a  given 
diameter  and  condition  of  pipe,  but  this  condition,  and  even  the 


18  PRACTICAL   HYDRAULIC   FORMULA. 

diameter  of  the  pipe,  is  liable  to  undergo  considerable  variation 
in  the  same  pipe  in  the  course  of  a  few  years. 

TABLE  OF  COEFFICIKNT3. 

Diameter  in  Value  of  Cfor  Value  of  C  for 

inches.  rough  pipes.  smooth  pipes. 

3  0.00080  0.00010 

4  0.00076  0.00038 
6                      0.00072  0.00036 
8                      0.00068  0.00034 

10  0.0006S  0.00033 

12  0.00066  0.00033 

14  0.00065  O.OOOS25 

16  0.00064  O.OOT32 

24  0.00064  0.00032 

30  0.00063  0.000315 

36  0.00062  0.00031 

48  0.00062  0.00031 

In  all  the  following  calculations,  the  coefficient  for  rough 
pipes  will  be  used. 

The  two  fundamental  equations  relating  to  the  flow  of  watefr 
through  long  pipes  are  : 

DX  H 

—  —  =  cr'  (i) 

L 

Q       =AV  (2) 

Equation  N~o.  2  will  generally  be  written  : 


,3, 


by  taking  the  value  of  Ffrom  (1). 

The  first  of  these  has  been   established  by  DARGY  ;  the  sec- 
ond is  based  upon  a  self-evident  proposition. 

In  these  equations  : 

D  -  diameter  of  pipe  ia  feet 

H  =  total  head 

L  =  length  of  pipe       "     " 

C  =  coefficient 

V  =  mean  velocity  in  feet  per  second 

Q  —  discharge  in  cubic  feet  per  second 

A  =  area  of  pipe  in  square  feet  =  D2  x  0.785 

TJie  above  two  formula  solve,  directly  or  indirectly,  all  prob- 


PRACTI  'AL   HYDRAULIC   FORMULA  19 

/.V  relating  to  the  flow  through  lung  pipe*,  and  all  such  prob- 
lem* muxt  be  brought  into  a  form  admitting  of  their  application, 
in  order  to  obtain  a  solution. 

H 

It  will  be   observed   that  —  is  the  rise  or  fall  per  foot  of 

Lt 

length  of  pipe,  and  is  therefore  the  natural  sine  of  the  inclina- 
tion of   the   slope  to  the  horizon.     This  relation  is  frequently 

H 

used  under  the  form  /  =  — .     Using  this  notation,  (1)  would  be 

L 
written  : 

D  i  =  c  r* 

In  long  pipes  the  length  is  generally  taken  as  being  equal  to 
the  horizontal  distance  separating  the  two  ends  of  the  pipe,  as  the 
difference  between  this  distance  and  the  actual  length  of  the  pipe 
is  relatively  insignificant.  If,  however,  a  case  should  present  it- 
self in  which  this  difference  was  considerable,  the  actual  length 
of  pipe  should  be  taken.  Further  on,  an  extreme  case  of  this 
kind  will  be  given,  presenting  isome  interesting  features. 

Some  practical  examples  of  the  use  of  these  formulae  will  now 
be  given.  In  all  that  follows,  the  resistances  of  entry,  exit,  and 
velocity  will  be  neglected,  and  the  total  head  will  be  considered  as 
available  for  overcoming  friction.  The  examination  of  cases  where 
the  above  factors  are  included  is  reserved  for  a  later  portion  of 
this  paper,  as  they  are  of  secondary  importance  when  dealing  with 
long  pipes. 

Example  1. — A  pipe,  1  ft.  in  diameter  and  1,000  ft.  long,  has 
a  total  fall  of  10  ft.  What  are  the  velocity  and  volume  of  its  dis- 
charge ? 

Substituting  the  given  values  in  (I)  we  have  : 
i  x  10 

=  0.00066  V* 

1000 

•  V  =  3.39  ft.  per  second. 


20  PRACTICAL   HYDRAULIC   FORMULA. 

Using  this  value  of  Fin  (2),  we  Lave  : 

Q  =  0.785  X  3.89 

Q  =  3.065  cu  ft.  per  second. 

Example  2.  —  Two  reservoirs,  having  a  difference  of  level  of 
water  surface  of  30  ft.,  are  joined  by  a  pipe  3,000  ft.  long.  What 
should  be  the  diameter  of  the  pipe  to  deliver  16  cu.  ft.  of  water 
per  second  from  the  upper  to  the  lower  reservoir  ? 

Eliminating  V  between  (1)  and  (2)  we  have  : 

D  X  H        Q* 

~LY~C~  ~A*' 
Observing  that  A  —  D2  0.785  ; 

J)X  H  Q2 


_ 

L  X   C  ~  D*  X  0.616 

Whence 

D*  -  Q!^_LX_C 

H  X  0.616 

If  we  knew  the  proper  value  of  the  coefficient  C  in  the  above 
equation,  it  could  be  immediately  solved,  and  the  value  of  D  ob- 
tained. But  C  varies  with  the  diameter,  and  the  diameter,  is  as 
yet  unknown.  We  must  therefore  have  recourse  to  "  Trial  and 
Error"  for  a  solution. 

Suppose  it  should  appear  to  us,  at  first  sight,  that  a  12-in. 
pipe  was  likely  to  be  of  the  proper  size.  We  therefore  take  C  = 
0.00066,  and  write  : 

_  256  X  3000  X  0.00066 

30  X  0.616  " 
D*  =  27.70 
D  =  1.94  ft. 

From  this  we  see  that  the  pipe  should  be  nearly  2  ft.  in  diam- 
eter, and  as  we  have  taken  too  large  a  coefficient  (that  for  24  ins. 
=  0.00064),  we  are  sure  that  1.94  is  too  large.  As  pipes  are  never- 
made  of  fractional  diameters,  the  above  value  of/)  would  be  taken 
=  24  ins.,  and  therefore  we  would  push  the  calculation  no  further. 
If  the  case  had  happened  to  be  one  requiring  minute  accuracy,  we 
would  recalculate  the  above  equation,  using  0.00064  for  the  value 


PRACTICAL   HYDRAULIC   FORMULA.  21 

of  C.  The  result  would  be,  D  =  1.93  ft.  nearly,  practically  the 
same  as  the  value  already  obtained. 

The  above  examples  (which  are  those  commonly  occurring  in 
practice)  are  very  simple,  and  involve  only  the  direct  application 
of  the  fundamental  formulae.  Let  us  now  consider  cases  of  a  more 
complicated  character,  where  they  can  only  be  used  indirectly,  and 
where  a  certain  amount  of  judgment  and  tact  is  required  in  the 
preparation  of  the  data. 

Example  3. — Suppose  a  reservoir  R  (Fig.  4)  containing  a  depth 
of  water  of  50  ft.  above  the  center  of  the  horizontal  pipe  A,  1  ft. 
in  diameter  and  1,000  ft.  long,  connected  by  a  reducer  with 
another  horizontal  pipe  B,  2  ft.  in  diameter  and  3,000  ft.  Jong. 
It  is  required  to  calculate  the  piezometric  head  li  at  the  junction, 
from  which  the  discharge  can  be  calculated,  and  the  hydraulic 
.grade  line  abc  established. 


\ 

\ 
\ 
\ 


1000  3000 

FIG.  i. 


It  is  evident  that  the  24-in.  pipe  must,  under  the  head  h,  dis- 
<i',.arge  the  same  quantity  per  second  as  the  12-in.  pipe,  under 
t\,.,3  head  50  —  7i.  We  have  then  from  (3)  the  equality  : 


I."  j/j 


2.X  h         _  n  7RS        ljCjaO^-A) 
300U  X  0.00064  \  1000  X  0.00066* 


Dividing  by  0.785,  squaring,  and  simplifying  : 


0.02          0.22 

•whence 

h  =  4.17. 


We  can  now  very  readily  get  the  discharge,   by  substituting 


22  PRACTICAL   HYDRAULIC  FORMULAE. 

the  value  4.17  for  h  in  either  member  of  the  above  equality. 
Thus  : 


Q  =  3.14  |/~  -  6.54  cu.  ft.  per  sec. 


Verifying  in  the  other  member — a  precaution  which  should 
never  be  neglected — \ve  obtain  the  same  result. 

It  is  evident  that  the  diameter  of  B  may  be  assumed  so  large 
that  no  value  of  h  can  be  found  to  satisfy  the  condition  that  both 
pipes  shall  run  full  with  the  given  height  of  water  in  the  reser- 
voir. In  such  a  case  the  pipe  B  serves  only  as  a  trough  to  re- 
ceive the  water  discharged  through  A  under  a  head  of  50  ft. 

Suppose  that  in  the  above  example  the  places  of  the  two 
pipes,  A  and  B,  should  be  changed.  Evidently  we  should  have: 

h  =  45. 8d. 

This  piezometric  height  would  give,  with  the  transposed 
position  of  the  pipes,  the  same  discharge  as  before,  the  only  dif- 
ference being  a  notable  change  in  the  hydraulic  grade  line.  If 
the  pipes  were  tapped  by  brandies,  the  greater  elevation  of  the 
grade  line  in  this  case  would  bring  a  much  greater  pressure  upon 
the  branches,  enabling  them  to  deliver  water  at  a  higher  level 
than  in  the  first  position  of  the  pipes. 


If 

•A83S 


5QO         800  1400  600 

FIG.  5. 

The  above  example  may  be  extended  so  as  to  cover  cases 
where  pipes  of  several  different  diameters  are  used.  Thus,  sup- 
pose a  system  of  pipes,  such  as  is  shown  in  Fig.  5,  where  a  res- 
ervoir with  a  head  of  50  ft.  of  water,  as  before,  is  tapped  by  ;i 
horizontal  line  of  pipes,  consisting  in  order  of  500  ft.  of  12-in.; 
800  ft.  of  16-in.,  1,400  ft.  of  8-in.,  and  600  ft.  of  6-in.  pipe. 


PRACTICAL   HYDRAULIC   FORMULA.  23 

This  example  may  be  worked  in  the  same  way  as  the  previous 
one,  by  getting  equations  for  h,  h',  and  h"  expressed,  by  substitu- 
tion, in  terms  of  h.  But  it  will  be  easier  to  treat  the  question  in 
another  way,  which  will  also  exhibit  the  further  resources  which 
we  have  at  our  disposal  in  solving  hydraulic  problems. 

Since  each  section  of  pipe  must  discharge  equal  volumes  in 
equal  times,  it  is  evident  that  the  respective  velocities  of  flow 
must  vary  inversely  as  the  areas  of  the  pipes.  These  areas  vary 
as  the  squares  of  the  different  diameters.  Designating,  therefore, 
by  Fthe  lowest  rate  of  velocity,  i.  e.,  that  of  the  water  passing 
through  the  largest  pipe  (the  16-in.  one),  we  obtain  the  relative 
velocities  in  the  other  pipes  by  multiplying  V  by  the  ratio  of  the 
square  of  the  diameter  of  the  largest  pipe  to  the  squares  of  the 
other  diameters.  It  will  be  convenient  to  form  the  following 
table: 


Lengths  in  ft. 

Diameters  in  ft. 

Velocities  in  ft. 
per  second. 

500 
800 
1,400 
600 

1 

1.78  V 
V 
4  V 
7.11  V 

Beginning  at  the  lower  end  of  the  system,  that  is  with  the 
6-in.  pipe,  and  employing  formula  (1)  in  which  h  and  Fare  the 
unknown  quantities,  we  have: 

1  /i 

—  X =  0.00072  X  (7. ID2  X  F2; 

2  600 

whence  :  h  =  43.68  F2 

2       (h'-h)      2       /A' -43.68  F2  x 

again:         ixi^-=iH— i^—) 

whence  :  h'  =  66  86  F2 

4       (h"  —  h')        4       /fc"- 66.86  F2\ 
8imilarly  :  _  x   __  _  =  ?  x  ^ __)  =  O.OU065  X 

whence  :  A"  =  67.25  F2 

50  -  h"      50  -  67.25  F2 

Finally  :  = =  0.00066  X  (1.78)2  X  F2 

500  500 

whence  :  F*  =  0.7321 

F  =  0.8556  ft.  per  second. 


24  PRACTICAL   HYDRAULIC   FORMULA. 

Substituting  this  value  of  F-  in  the  above  equations  ; 

h  =  31.98  ft. 
h'  =  18.95  " 
h"  =  49.23  " 

We  also  get  the  velocities  in  the  different  pipes,  thus  : 


6  inch,  velocity  =  7.11  X  0.85*5  = 

8  "  *'  =  4  X  0.856  —  3.424 

16  "  "  =  1  X  0.856  =  0.856 

12  "  "  =  1.78  X  0  856  =  1.524 

The  work  can  be  checked  by  using  the  above  values  of  h,  h'. 
and  A",along  with  the  other  data,  in  (1),  and  obtaining  the  veloci- 
ties in  this  way. 

Thus,  beginning  with  the  6-in.  pipe  : 

1  31.98 

—  X  -  =  0.00072  V- 

2  600 

V  =  6.08 

2  16.97 

—  X  --  =  0.00069  V" 

3  1400 

V  =  342 

4  0.28 

—  X  --  =  0.00065  V"* 
3    800 

V"  =  0.85 
0.77 

1  X  --  =  0.00065  F'"2 
500 

V"  =  1.53 

A  very  close  agreement  throughout. 

In  the  above  calculations  the  decimals  have  been  carried  out 
further  than  would  ordinarily  be  necessary  in  practice.  It  was 
done  in  the  present  instance  in  o'rder  to  avoid  discrepancies  in 
checking. 

We  have  another  check,  in  the  volumes  discharged.  Thus 
the  discharge  through  the  6-in.  pipe,  with  the  given  velocity,  is 
by  (2): 

Q  =  0.195  X  6.086 

Q  =  1.19  cubic  ft.  per  second. 

All  the  other  pipes  should  have  an  equal  discharge;  for 
instance,  the  12-in.  pipe  gives  : 


=  0,78  X  1.524 

=  1.1U  cubic  ft.  per  second. 


CHAPTER  II. 

Calculations  are  the  Same  for  Pipes  laid  Horizontally  or  on  a  Slope—Qualification 
of  this  Statement— Pipe  of  Uniform  Diameter  Equivalent  to  Compound  System 
— General  Formula — Numerical  Example— Use  of  Logarithms  (foot  note} — Nu- 
merical example  of  branch  pipe -Simplified  method— Numerical  Examples— 
Relative  discharges  through  branches  variously  placed — Discharges  determined 
by  plotting— Caution  regarding  results  obtained  by  calculation— Numerical 
examples. 

In  the  preceding  examples  a  series  of  horizontal  pipes  has 
been 'considered,  the  head  being  produced  by  an  elevated  reservoir 
placed  at  one  end.  The  results  would  have  been  identical,  how- 
ever, if  the  head  had  been  produced  by  the  pipes  being  laid  upon 
a  slope,  provided  the  difference  of  level  between  the  two  extremi- 
ties remained  the  same,  for  the  velocities  and  hydraulic  grade  line 
would  remain  unaltered.  The  pressure  in  the  pipes  would  vary 
however,  according  to  their  distance  below  the  hydraulic  grade 
line,  the  pressure  being  measured  at  any  given  point  in  the  pipe 
line,  by  the  vertical  distance  between  such  point  and  the  grade 
line.  If  the  pipes  were  laid  exactly  upon  the  hydraulic  grade  line 
there  would  be  no  pressure  at  all  in  the  pipes,  and  if  they  rose  at 
any  point  above  it,  there  would  be  either  no  flow  or  a  diminished 
one,  unless  siphonage  were  resorted  to. 

In  order  to  make  this  point  very  plain,  we  will  consider  the 
same  system  of  pipes  as  that  used  in  the  last  example,  but  laid  as 
shown  in  Fig.  6.  the  upper  extremity  being  fed  by  a  constant 
supply,  with  only  head  enough  to  overcome  resistance  to  entry, 
and  produce  initial  velocity,  which  will  be  treated  of  further  on 

Calculating  precisely  as  before,  we  get   the   same,  hydraulic 


26 


PilACriCAL   HYDRAULIC    FORMULA. 


grade  line,  unbroken  by  the  rising  grade  of  the  last  200  ft.    of 
6-in.  pipe. 


200 


FIG  6. 


It  is  sometimes  desirable  to  ascertain  the  uniform  diameter  of 
a  pipe  which  shall  be  equivalent  to  a  series  of  pipes  of  different 
diameters,  such  as  wo  have  just  been  studying.  This  may  be 
done  by  an  application  of  formula  (4),  which  for  this  purpose  is 
written  in  the  following  form: 

C  Q*        L 
H=- 

0.616       D* 

As  an  example,  let  us  calculate  the  diameter  of  a  single  pipe, 
of  the  same  total  length  and  fall  as  the  series  of  pipes  which  we 
have  just  had  under  consideration,  and  capable  of  discharging 
an  equal  volume.  We  will  first  establish  the  general  formula  for 
all  such  problems,  expressing  the  diiference  of  piezornetric  level 
between  the  two  ends  of  each  pipe  respectively,  by  A1,//.g,7i3,//4, 
etc.,  their  respective  lengths  by  l^il^l^l^  etc.,  their  respective 
diameters  by  d^d^d^d^,  etc.,  and  their  respective  coefficients  by 
Cic2csc4»  etc.,  commencing  with  the  lower  end.  We  will  express 
the  total  length  by  L,  the  total  difference  of  level  by  H,  the  un- 
known diameter  by  D,  and  its  coefficient  by  C. 

Now,  observing  that  the  quantity  discharged  per  second  by 
each  pipe  is  the  same,  we  have  the  4  equations: 

c,   Q*       I, 

h,  =  -  x  — 

0.616        d% 


0.616 


PRACTICAL   HYDRAULIC   FORMULAE.  27 

_  c,   Q*       Zs 
0.616      di 

C4     Q"*  14 

0.616      df 

Adding,  and   observing   that   the  sum  of  the  partial  heads 
hl  h2  h  3  h±  equals  H,  we  have  : 

H  - —  (— -+-^-4-  £LJ_  +  £I_I_\ 

0.616    \  d,5          da6          d3s          d46    ) 

but  we  have  also  the  equation 

CQ*       L 

H= X  — 

0.616       D6 

whence,  suppressing  the  common  factor  : 

C  L      c,  li      c2  Z2      ca  la      c4 14 

_—=-_.+  -_+ 1  +  1J  (5 

The  above  is  the  general  formula. 
Substituting  the  special  values  of  our  example  : 

3300  0.33       0.52        0.966         0.432 

—  X  C=  —  +  —-  +  -—-+__ 

Giving  a  preliminary  approximate  value  to  C  of  0.00066,  we 
have 

2.178 
— -—  =  0.33  +  0.123  +  7.335  -f  13.821 

iy>  =  0.1007 

D    =  0.63 

This  value  of  D  indicates  a  practical  diameter  of  8  ins. 
In  order  to  check  this  value,  we  may  write  (4)  under  the 
form  : 

o  =    /D*  x  H  x  °-6lS 

V      \  Lx  C 

Substituting  given  values  : 

1'|Q7  X  50  X  0.616 
2.178 

Q  =  1.193  cu.  ft.  per  second, 

thus  proving  the  correctness  of  the  work. 


28  PRACTICAL    HYDRAULIC    FORMULA. 

These  calculations  can  be  abridged,  and,  in  many  cases,  suffi- 
cient accuracy  secured  by  adopting  a  mean  common  value  for  (7. 
If  we  do  so  in  the  present  case,  0  becomes  a  common  factor,  and 
disappears  from  the  calculation,  (5)  becoming 

L        I,         I,         13 

-  + 1--—,  etc.  (5)  bis 

DB     as   as    d3* 

If  this  equation  be  worked  out  for  the  above  given  values,  we 
have  : 

D  =  0.64 

or  8  ins.  as  before. 

It  will  be  observed  that  this  process  might  have  been  used 
with  advantage  in  the  previous  example,  by  ascertaining  the  dis- 
charge of  an  equivalent  pipe,  and  then  calculating  the  heads 
necessary  to  produce  this  discharge  through  the  different  pipes. 

In  calculating  fifth  powers  and  roots,  a  table  of  logarithms  is 
almost  indispensable.  If  none  is  at  hand  a  table  of  squares  and 
cubes  is  of  some  use,  remembering  that  a  number  can  be  raised  to 
the  fifth  power  by  multiplying  together  its  square  and  cube.  Fifth 
roots,  in  the  absence  of  logarithms,  can  only  be  extracted  by 
il  trial  and  error/'  using  the  above  rule  for  fifth  powers.* 

Example  4th.  A  horizontal  pipe  (Fig.  7),  48  ins.  in  diameter 
and  2,000  ft.  long,  issues  from  a  reservoir  in  which  the  surface  of 
the  water  is  maintained  at  a  constant  height  of  50  ft.  above  the 
center  of  the  pipe.  Midway,  this  pipe  is  tapped  by  a  branch  pipe 
24  ins.  in  diameter  and  500  ft.  long,  with  a  rising  grade  of  4  ft.  in  500. 
What  is  the  piezometric  head  li  at  the  junction,  and  what  the 
discharge  from  each  pipe  ?f 

It  is  evident  that  the  48-in.  pipe  above  the  junction  musts, 
with  the  head  50 — li,  discharge  as  much  water  per  second  as  the 

*  All  hydraulic  calculations  are  greatly  facilitated  by  the  use  of  logarithms;  and 
those  engaged  in  making  such  calculations  should  not  fail  to  familiarize  themselves 
with  the  us>e  of  the&e  powerful  auxiliaries  to  arithmetical  work. 

t  With  these  lengths  and  diameters,  the  ahove  system  does  not  properly  come 
under  the  classification  of  "long  pipes."  As  the  present  object  is  only  to  exemplify 
methods  of  calculation,  the  example  is  equally  good. 


PRACTICAL   HYDRAULIC   FORMULA.  g£ 

combined  discharge  of  the  48-in.  pipe  below  the  branch  with 
the  head  h,  and  the  24-in.  pipe  with  the  head  li— 4.     From  (3), 


FIG.  7. 


which  in  this  case  will  perhaps  be  the  most  convenient  equation 
for  quantity,  though  that  derived  from  (4)  is  frequently  useful, 
we  have  : 


1000  X  0.00062 

q  =  12.56  4/ L? 

1000  X  0.00062 


500  X  0.00064 

which,  put  in  equation,  give  : 


1000  X  0.00062 


12.56.4/ t?. 

r     innn  v  n  n 


+  3.U   •'      2(ft-4> 


1000  X  O.OC062  500  X  0.00064 

The  coefficients  0.00062  and  0.00064  are  so  nearly  equal  that 
we  may,  in  the  following  calculations,  discard  them  as  common 
factors.  Dividing  by  3.14  and  striking  out  also  the  common 
factors  -j-^oo  an(*  T!  o>  we  nave  simply  i 


4  Vso  -  h  =  4  I/A"  +  I/A—  4 

Squaring  800-16/i  =  16A  +  /i  —  4  +  8  |/A»  —  4  A 


which  gives:  33  A  =  804  -8  VA»  -  4  A 

Neglecting,  for  a  first  approximate  value  of  li  the  quantities 
affected  by  the  radical  : 


33A  =  804 


30  PRACTICAL   HYDRAULIC   FORMULAE. 

Neglecting  decimals : 

h  =  24. 

Substituting  this  value  for  h  under  the  radical  : 


33A  :=  804  -  8  V576  -  96 

\vliich  gives,  always  neglecting  decimals,  a  second  approximate 
value  : 

h  =  19. 

A  third  and  fourth  approximation  give  respectively  //  =  20.3 

and  h  =  20. 

We  will  take  20.1  as  very  near  the  true  value.* 
Substituting  20.1  in  place  of  h  in  the  equations  giving  the 

quantities  discharged,  we  have  : 


/4  x  29.9 

Q  =  12.56  A/ =  174.45 

r       n  ft* 


/4  X  20.1 

12.56  A/ =  143.05 

r       p.  R9. 


/2  X  16.1 

q'  =  3.14  |/ =  31.50 

"       0.32 

We  have  thus  :  Q  =  a  +  q'. 

The  ahove  method  gives  directly  the  true  value  of  h;  but  it 
involves  tedious  figuring,  even  in  our  example,  which  happens  to 
admit  of  many  simplifications  owing  to  the  number  of  common 
factors.  It  will  be  easier,  and  often  shorter,  to  obtain  the  value 
of  h  by  first  assuming  one  which  we  judge  likely  to  be  near  the 
truth,  calculating  what  discharge  it  would  give  from  the  two 
branches,  and  then  calculating  the  head  necessary  to  discharge 
the  same  quantity  from  the  single  pipe  above  the  branch.  Then, 
comparing  the  total  height  thus  obtained  with  the  known  height 
of  the  water  in  the  reservoir,  we  can  deduce  the  true  value  of  h 
by  a  proportion. 

Let  us  apply  this  method  to   the  above  example.     We  know 

*  The  value  of  h  may  he  obtained  directly  by  using  the  usual  formula  for 
adfected  quadratic?  ;  bur,  with  the  aid  of  a  table  of  squares  and  square  roots,  the 
above  approximate  method  will  generally  be  the  easier  and  quicker  one. 


PRACTICAL   HYDRAULIC   FORMULAE.  31 

at  once  that  li  must  be  less  than  25.  because  that  would  be  its 
value  if  the  24-in.  branch  were  closed.  Supposing  we  judged 
that  22  ft.  would  be  about  correct.  We  then  have  to  solve  the  two 
equations  : 


q  =  12.56  \/ =  149.60 

"         n  RO 


X  22 
0  62 


/  2  X  18 

q'  =--3.14   {  -  =  33.30 

0.32 

also,  for  the  equal  discharge  through   the  48-in.   pipe  above  the 
branch,  squaring  (3),  we  have  : 

(182.90)2  X  0.62 

h  = =  32.87 

(12. 56) 2  X  4 

This  height,  added  to  2*2,  the  assumed  value  of  h,  gives  a 
total  height  of  54.87  ft.  as  against  50  ft.,  the  actual  total  height. 
By  proportion  we  have  : 

h         50 


22       54.87 

This  value  of  h  agrees  with  that  already  found. 
If  the  24-in.  branch  were  closed  we  should  have  for  the  dis- 
charge : 


1.24 

When  the  24-in.  branch  was  open  we  had  a  total  discharge  of 
174.73  cu.  ft.  per  second.  There  is  an  increase,  therefore,  of 
about  9|  per  cent,  by  opening  the  branch. 

Let  us  now  see  what  the  discharge  would  be  if  the  branch 
were  placed  only  500  ft.  from  the  reservoir,  instead  of  1,000  ft., 
all  the  other  conditions  remaining  the  same. 

We  will  assume  h  =  33  ft.  and  solve  the  two  equations 

/     Tx~33 

q  =  12.56  4/  ---  =  149.5 
\     1.500  X  0.00062 


2  X  29 



0.32 


32  PRACTICAL   HYDRAULIC   FORMULAE. 

(191. 8)2  X  0.31 

also  h'  = 18.07 

(12  56,2  X  4 

giving  a  total  height  of  51.07  as  against  53.     Reducing  : 

h        50 

33  ~  51.07 
h  =  32.3 

Using  this  value,  instead  of  the  assumed  one,  we  have  : 


/  4  X  17.7  /4X323  /  2  X  23.3 

12.56 1/ 

r     0.31  r     0.93 


17.7  /4X323  /2X  23.3 

—  =  12561/ -Kllti/ 

I  r        0.93  f*      0.32 

189.83  =  148.03  +  41.76 

very  nearly. 

As  compared  with  the  discharge  when  the  24  in.  branch  is 
closed  this  shows  a  gain  of  19  per  cent.,  just  double  the  gain 
when  the  branch  was  located  at  the  center  of  the  pipe. 

Supposing  now  that  the  branch  were  placed  1,500  ft.  from  the 
reservoir.  Assuming  10  ft.  as  a  probable  value  of  h  we  have  : 

q  =  12.56   / - =  142.46 

'    500  X   0.00062 


/V  X  b 
137  = 


X6 

19.23 


(161.7)2  X  0.93 

also:  h'  = —  =  38.53 

(12.56)8  X   4 

h          50 
By  propor<ion  -  =  — 

h  =  10  30 

Using  this  value  instead  of  the  assumed  one  : 


/  4  X  39.7  /  4  X  10.3  /  2  X  ti.3 

12.56  A/ =  12.56  |/ +  3.14  I/ 

V        0.93  r         0.32  "         0.32 


164.13  =  144.574-19.68 


very  nearly. 

As  compared  with  the  discharge  when  the  24-in.  branch  is 
closed,  this  shows  a  gain  of  not  quite  3  per  cent,  which  is  in 
marked  contrast  to  the  gain  when  the  branch  was  only  500  ft. 


PRACTICAL   HYDRAULIC   FORMULAE.  33 

from  the  reservoir,,  being  less  than  one-sixth  of  the  gain,  in  that 
case. 

It  will  be  interesting  to  study  a  little  more  in  detail  the  ques- 
tion of  relative  discharges.  We  have  seen  that  when  there  is  no 
branch  open  on  the  48-in.  pipe,  its  discharge  is  159.51  cu.  ft. 
per  second.  The  24-in.  branches,  wherever  placed,  increase  the 
total  discharge,  but  diminish  that  in  the  48-in.  pipe,  below  the 
branch.  By  comparing  the  above  quantities,  it  will  be  perceived 
that  the  flow  from  the  48-in.  pipe  is  diminished  approximately 
by  that  proportion  of  the  quantity  flowing  through  the  24-in. 
branch  which  is  represented  by  its  proportionate  distance  from 
the  reservoir.  Thus,  when  the  branch  is  1,500  ft.,  or  three- 
quarters  of  the  length  of  the  48-in.  pipe,  from  the  reservoir,  as 
in  the  last  case,  its  discharge  is  19.62  en.  ft.  per  second.  Three- 
quarters  of  this  quantity  is  14. 715,  which,  subtracted  from  159  51, 
leaves  144.795,  or  very  nearly  that  of  the  48-in.  pipe  below  the 
branch,  as  determined  by  calculation. 

In  the  same  way  half  of  the  discharge,  when  the  branch  is 
situated  half  way  from  the  reservoir,  subtracted  from  159.51, 
gives  also  very  nearly  the  amount  discharged  below  the  branch. 
When  the  branch  is  500  ft.,  or  one-quarter  of  the  total  distance, 
from  the  reservoir,  one-quarter  of  its  discharge  taken  from  159.51 
gives  very  closely  the  discharge  as  calculated  for  the  48  in.  pipe 
below  the  branch. 

Let  us  now  take  an  extreme  position  for  the  branch,  and 
suppose  it  placed  close  to  the  reservoir,  so  that  there  is  practically 
no  portion  of  the  48-in.  pipe  between  it  and  the  reservoir.  There 
will,  therefore,  be  no  part  of  the  flow  from  the  branch  subtracted 
from  that  of  the  main  pipe,  and  the  two  will  each  discharge  the 
same  quantity  as  if  the  other  were  not  there.  That  is,  the  48-in. 
pipe  will  discharge  159.51,  and  the  24-in.  53.24  cu.  ft.  per  second. 

If  we  should  take  another  extreme  position  for  the  branch, 
and  suppose  it  placed  at  the  end  of  the  48-in.  pipe,  it  is  obvious 
that,  with  its  assumed  rising  grade  of  4  ft.  in  500,  it  would  dis- 


34 


PRACTICAL  HYDRAULIC   FORMULA. 


charge  no  water  at  all.  A  position  could  be  found  by  trial  where 
it  would  just  cease  to  discharge  water,  but  for  the  object  of  the 
present  investigation  this  is  not  necessary. 


FIG.  8. 

If  the  above  results  are  plotted,  as  in  Fig.  8,  a  very  instruc- 
tive diagram  is  obtained.  The  successive  500  ft.  lengths  being 
laid  off  as  abscissae,  and  the  discharges  measured  upon  the  corre- 
sponding ordinates,  it  will  be  seen  that  their  extremities  all  lie 
nearly  in  the  same  straight  line.  If,  therefore,  the  discharges  for 
any  two  positions  of  the  branch  be  calculated,  and  the  straight 
line  drawn  passing  through  their  extremities,  the  discharge  for 
any  other  position  of  the  branch  can  be  obtained  by  erecting  an 
ordinate  at  the  given  point  to  the  straight  line,  and  the  flow 
through  the  main  also  obtained  by  subtracting  the  proper  portion 
of  that  of  the  branch. 

In  practice,  when  making  calculations  si  mils;'  to  those  under 
consideration,  one  error  must  be  carefully  guarded  against, 
namely,  the  supposing  that  the  actual  results  will  be  exactly  tw 
calculated.  The  chief  value  of  these  calculations  lies  in  the  facfc 
that  they  furnish  pretty  trustworthy  relative  results,  that  is,  they 
establish  fairly  well  in  practice  the  fact  that  if  a  certain  pipe  de- 
livers a  certain  volume  of  water  in  a  certain  position,  it  will  de- 
liver a  certain  greater  or  less  amount  in  another.  The  actual 
amounts,  in  either  case,  cannot  be  surely  determined,  as  they  de- 


PRACTICAL   HYDRAULIC   FORMULAE.  33 

pend  upon  ^  many  varying  circumstances  about  which,  even  when 
aware  of  th<r  existence,  we  liave  no  exact  data. 

Let  us  next  suppose  a  system  in  which  the  48-in.  pipe  is 
tapped  everyiOO  ft.  by  a  24  in.  pipe,  500  ft.  long,  laid  as  before 
with  a  grade  'f  4  ft.  in  500. 

Assumip  a  height  of  9  ft.  for  the  piezometric  column  li 
nearest  the  fie  end  of  the  pipe  we  have  : 

X  9  t  /2  X  5  /  4  (h  —  9) 


t  / 
.56  V 


t  /2  X  5 

1-  3.  Hi/  --  =  12.56  \/ 
0.32 


0.31  0.32  0.31 

Since  th  'denominators  under  the  radicals  are  so  nearly  equal 
we  may  cancc  them,  and  making  other  simplifications,  write  : 


Whence: 
Again: 

Also 


By  proportion  we  ive:  —    =  -- 

9  63.97 
h  =  7.05 

As  the  vaie  of  h'"  =  63.79  differs  considerably  from  the 
true  value  ==  5,  and  as  the  above  proportion  is  not  exactly  abso- 
lute, particular  in  a  somewhat  complex  system  like  the  present, 
it  is  probable  tat  the  value  just  obtained  for  It  is  not  a  sufficiently 
close  approximtion  to  answer  our  purpose.  We  will  therefore 
make  a  second  nlculation,  using  7  as  a  second  approximate  value 
for  //. 

Carrying  to  calculation  through  precisely  as  above,  we  obtain 
the  following'  vines  : 

h  =  7.32 
h'  =  16.12 
h"  =  29.60 
h"'  =  50.00 


36  PRACTICAL  HYDRAULIC  FORMULAE. 

Calculating  the  various  discharges  under  these  piezometric- 
heads,  calling  those  through  the  different  sections  of  48-in.  pipe, 
commencing  at  the  lower  end,  Q,  Q' ,  Q",  Q'",  and  those  through 
tire  corresponding  24-in.  branches,  q,  q',  q",  we  have  : 

Q     =  122.05 
q      =1430 

Q  +  q  =  136.35 
Q'  =  136.10 
q'  =  27.62 

Q'  +q'  =  163.72 
Q"  =  163.75 
q"  =  39.72 

Q"  +  q"  =  203.47 
Q'"  =  203.75 

These  results  show  a  very  close  agreement. 

It  is  worthy  of  note  that  the  total  discharge  in  this  case  is 
not  greatly  increased  over  that  obtained  with  a  single  branch  sit- 
uated 500  feet  from  the  reservoir.  In  general  it  will  be  found,  as 
in  these  two  cases,  that  when  a  main  is  tapped  at  a  certain  point 
by  a  single  branch,  the  total  discharge  is  comparatively  but 
slightly  increased  by  the  introduction  of  a  series  of  similar  branches 
placed  below  the  first  junction.  The  position  of  the  first  branch, 
however,  has,  as  the  above  examples  show,  a  very  great  influence 
both  on  the  volume  of  discharge  and  the  form  of  the  hydraulic 
grade  line.  This  latter  feature  merits  careful  attention. 

It  will  be  interesting  to  study  the  effect  upon  the  flow  through 
such  a  system  as  we  have  been  just  considering,  when  the  condi- 
tions are  somewhat  changed.  For  instance,  in  the  last  example 
let  us  suppose  that  the  three  branch  pipes,  instead  of  having  each 
an  equal  rising  grade  of  4  feet  in  their  length  of  500  feet,  have 
rising  grades  respectively  of  4  feet,  12  feet  and  24  feet  in  500, 
commencing  at  the  lower  branch,  all  other  conditions  remaining 
the  same. 

Assuming,  as  before,  an  approximate  value  for  h  of  9  feet,  we 
get,  as  before 

h'  =  20.52 


PRACTICAL   HYDRAULIC   FORMULAE.  37 

Our  next  equation  will  be  :, 

—    \/17.04  =     \/  h"  —  20.52 


h"  =  35.81 

Again  : 


23.62=    |»'_35.81 
h'"  =  56.24 

This  value  is  sufficiently  near  the  given  one  of  50,  to  warrant 
our  using  it  to  obtain  pretty  close  approximate  values,  by  propor- 
tion, as  follows  : 

h  =  8  00 
h'  =  18.24 
h"  =  31.83 

h'"  =  50.00 

Whence  we  obtain  the  following  discharges 


/    32 

Q  =  12  56  A/  -  =  127.6 
V     0.31 


q  =    3.U  A  /  -  =    15.7 
V     0.32       - 


=  143.3 


12.48 
Q'=    3."V-^T=    196 

Q'-t  q'  =  164.0 
"5T36 


0.21 


Q"  =  12.56  A/ 


/  15.66 

q"  =    3.14  A/-    —  =    22. 
V     o.32      


0 
=  188.3 


72.68 

=  T.J.56  \/  -     -4-  192.3 
0.31 


This  shows  a  pretty  fair  agreement  between  the  volumes  dis- 
charged, the  discrepancies  being  due  to  the  fact  that  our  assumed 


38  PRACTICAL   HYDRAULIC   FORMULAE. 

value  of  h  was  not  sufficiently  close  for  a  line  calculation.  The 
figures  are  near  enough,  however,  to  serve  the  purpose  of  showing 
to  how  small  an  extent,  comparatively,  the  results  are  changed  by 
the  very  considerable  changes  made  in  the  inclination  of  the 
branch  pipes.  Later  on  we  shall  have  occasion  to  notice  more 
fully  the  small  relative  changes  made  in  the  volumes  discharged 
through  given  pipes  by  changes  of  grade:  for  the  present  we  will 
only  call  attention  to  the  slight  variations  produced  in  the  hy- 
draulic grade  line,  as  determined  by  the  piezometric  heads. 


CHAPTER  III. 

Numerical  example  of  a  system  of  pipes  for  the  supply  of  a  town— Establishment 
of  additional  formulce  for  facilitating  such  calculations-  Determinations  of 
diameters— Pumping  and  reservoirs — Caution  regarding  calculated  results — 
Useful  approximate  formulce— Table  of  5th  powers— Preponderating  influence 
of  diameter  over  grade  illuftrated  by  example— Maximum  velocities.  (Note.) 

As  a  further  study  of  a  system  of  pipes  to  deliver  water,  let 
us  suppose  a  town  divided  by  intersecting  streets  into  blocks  1,000 
ft.  sq.,  as. shown  in  Fig.  9.  We  will  suppose  that  the  proposed 
water  supply  requires  a  total  volume  of  3  cu.  ft.  per  second,  equal 
to  say  800,000  U.  S.  galls,  in  10  hours. 

The  water  is  to  be  introduced  by  a  central  main  ABC,  and 
delivered  east  and  west  by  the  side  mains  D  D',  E  E ',  FF', 
G  G',  H  H'.  At  the  extremities  of  these  mains,  the  water  is  to 
be  delivered  at  the  elevations  above  datum  indicated  by  the 
figures  placed  in  brackets.  The  side  mains  D  D'  and  E  E'  are  to 
deliver  each,  east  and  west,  J  cu.  ft.  per  second,  which  quantity 
we  will  suppose  is  to  be  carried  through  the  whole  length  of  the 
pipe  and  delivered  at  its  extremity  at  the  maximum  elevation, 
without  regard  to  the  quantities  drawn  often  route  by  the  service 
pipes  and  smaller  north  and  south  mains,  nor  those  drawn  off  by 
the  lower  taps.  This  will  secure  a  good  delivery  of  water  in  case  of 
lires.  The  total  delivery  of  the  above  two  side  mains  will  there- 
fore be  1  cu.  ft.  per  second.  The  remaining  three  side  mains, 
F  F' y  G  G' ,  and  ////',  are  to  deliver,  similarly,  J  cu.  ft.  per 
second  at  each  extremity,  making  2  en.  ft.  for  the  three. 

These  being  the  data,  we  will  suppose  the  problem  to  be  the 


40 


PRACTICAL   HYDRAULIC   FORMULAE. 


determining  of  the  respective  diameters  of  the  pipes,  and  the 
height  to  which  the  water  must  be  raised  in  a  supply  reservoir 
or  standpipe,  situated  somewhere  to  the  north  of  the  town. 


(185) 

D        5° 

* 
5"       A 

3        5" 

5"      D' 

(170) 

(180) 

k 

£         5" 

(201) 
5" 

2k        5" 

k 
5"      £' 

(165) 

1170) 

i 

F          6" 

6"       B 

2         5" 

k 
5"     F1 

(ISO} 

(165) 

0       e" 

(186) 
6"    (181) 

5n 

5"      0' 

(145) 

tlW) 

H        6" 

a 

6"        C 

(176)      5" 

5"     H' 

(140) 

* 

% 

h 

FIG.  9a. 


The  problem  thus  stated  is  indeterminate  and  admits  of  an 
itidefinite  number  of  solutions,  for  we  may  either  use  large 
pipes  and  low  elevations,  or  small  pipes  and  high  elevations. 
Practically,  however,  there  are  limitations  to  this ;  for  in  the 
first  place  we  shall  naturally  be  restricted  as  to  the  height  to 


PRACTICAL    HYDRAULIC    FORMULAE.  41 

•which  it  would  be  possible  or  advisable  to  raise  the  water,  and 
secondly,  experience  shows  that  we  should  confine  ourselves 
within  certain  limits  as  regards  the  velocity  of  the  water  in  the 
pipes. 

Generally  speaking,  these  velocities  should  not  exceed  such 
as  would  be  produced  by  a  fall  of  from  4  to  8  ft.  per  thousand, 
according  to  the  size  of  the  pipe  ;  the  greater  fall  belonging  to 
the  smaller  diameter.  (See  note  at  end  of  chapter.) 

Before  commencing  the  calculations,  it  will  be  well  to  estab- 
lish certain  additional  formulae,  derived  from  (4),  which  are 
frequently  of  considerable  utility. 

When  the  length  and  diameter  are  constant  : 

Q'*  _  H' 
Q2       H 

When  the  head  and  diameter  are  constant  : 


When  the  head  and  length  are  constant  : 

Q'*       D'5  C 

Q-  ~D*  C' 


When  the  head  and  discharge  are  constant  : 

D'6      L'  C' 
~D*~    LC 

When  the  length  and  discharge  are  constant  : 

D5      HC' 
D*  ~  H  C 

These  relations  indicate  that,  other  things  being  equal,  the 
squares  of  the  discharges  vary  directly  as  the  heads  and  the  fifth 
powers  of  the  diameters,  and  inversely  as  the  lengths  ;  and  that, 
other  things  being  equal,  the  fifth  powers  of  the  diameters  vary 
directly  as  the  squares  of  the  discharges  and  the  lengths,  and  in- 
•versely  as  the  heads. 


42  PRACTICAL  HYDRAULIC  FORMULAE. 

As  these  relations  are  generally  used  for  approximations,  the 
coefficients  may  be  dropped,  and  the  equations  written  in  this 
form  : 

«-yv** 

'  n?  v  fT 

(7) 


(10) 


&  X  H 

iT 

Other  combinations  can  be  made  from  these  relations.  Thus  : 


X  HX 


Commencing  now  with  the  west  side  of  the  main  H  H',  we 
have  J  cu.  ft.  to  be  delivered  at  an  elevation  of  (160)  above  datum. 
As  the  pipe  will  be  a  comparatively  small  one,  we  will  assume  a 
grade  of  1()%o>  which  will  give  a  rise  of  16  ft.  between  the  ex- 
tremity and  the  main  junction,  and  requires  an  elevation  of 
piezometric  head,  at  this  junction,  of  (176),  as  shown  in  the  fig- 
ure. 

To  obtain  the  proper  diameter  of  pipe  for  this  grade  and  dis- 
charge, we  have,  using  (4),  and  assuming  C  =  0.00076  as  a  prob- 
able value ; 


A   /(^)2  X    1000  X  i 

=  \/  - 


X   LOOO  X  0.00076 
8  X  0.61 

whence  D5  =  0.017304 
and  D  =  0.444. 


PRACTICAL   HYDRAULIC   FORMULA.  43 

Or,  for  the  next  highest  even  inch  : 

D  =  6  inches. 

As  regards  the  diameter  of  the  pipe  on  the  east  side,  since 
the  length  and  discharge  are  the  same  as  for  the  west  side,  and 
only  the  heads  vary,  being  respectively  16  and  36  ft.,  it  can  be  ob- 
tained by  means  of  (11). 

Thus  : 

*  ,0017304  X16 
D'  = 


96 

D'  =  0.3777 

or,  for  next  highest  even  inch  : 

D'  =  5  inches. 

The  above  head  of  18  ft.  per  thousand  produces  a  velocity  of 
flow  in  a  5  in.  pipe  of  a  little  over  3  ft.  per  second,  which  is 
somewhat  greater  than  it  should  be.  If  the  limit  of  velocity  is 
overstepped  to  any  considerable  degree  in  a  system  of  pipes  such 
as  we  are  considering,  it  would  be  best  to  use  a  larger  pipe  and 
check  its  flow  down  to  the  desired  delivery  by  means  of  a  gate  or 
stop  cock  placed  near  its  upper  end,  the  effect  of  which  will  be  to 
diminish  the  head.  In  the  present  instance  the  excess  of  velocity 
is  probably  not  sufficient  to  render  this  precaution  necessary. 

The  elevations  are  such  that  the  above  diameters  of  6  and  5 
ins.  are  also  proper  for  the  side  mains  G  G',  F  F'. 

It  is  now  necessary  to  calculate  the  diameter  of  the  central 
main  from  B  to  C.  This  main  might  be  divided  into  two  parts, 
that  between  F  F'  and  G  G'  and  that  between  G  G'  and  H  H', 
but  we  will  calculate  it  upon  the  supposition  of  a  uniform  diameter, 
capable  of  delivering  the  entire  volume  of  f  cu.  ft.  per  second  as 
far  as  IT  H'. 

Assuming  a  probable  value  of  C  =  0.00066,  we  have  from  (4): 

16 

Z>5  =  -  X  1.32 
9 


44  PRACTICAL   HYDRAULIC   FORMULA. 

whence: 

D*  =  0.3817 

and: 

D    =  0.826  =  10  ins. 

Taking  now  the  mains  E  E'  and  D  D ',  and  beginning  on  the 
west  side,  assuming  as  before  a  grade  of  8  ft.  per  1,000,  we  find 
the  length  and  head  equal  to  those  of  F  F'  etc.,  the  only  differ- 
ence being  the  quantity  it  is  desired  to  deliver,  which  is  now 
i  cu.  ft.  as  against  £  in  F  F'.  The  relation  (9)  is  therefore  ap- 
plicable, and  we  have: 


z>=\/o. 


017304  X  - 
16 


1 

whence: 

I)'  =  0.0097335 

and 

D'  =  0.396 

or,  say, 

D'  =  5  ins. 

The  mains  on  the  east  side  are  determined  as  before: 


•/ 


16 

0.0097335  X  - 


D'  =  0.346 

This  is  not  quite  4J  ins.,  but  to  insure  the  desired  delivery, 
it  will  be  best  to  take  the  next  highest  even  inch,  and  call  it  5 
ins. 

As  regards  the  central  main  from  A  toB,  we  find  two  grades, 
the  upper  one  ^-f^  and  the  lower  J^-Q.  The  lower  section  must 
deliver,  under  a  grade  of  ^o,  all  the  water  required  for  F  F',  G 
G',  and  H  H',  aggregating  2  cu.  ft.  per  second.  Using  (4),  and 
taking  0.00066  as  a  probable  value  of  C,  we  have  : 

4  X0.66 
jys  -  -- 

6.1 


PRACTICAL   HYDRAULIC   FORMULAE.  45 

whence : 

Z>5  =  0. 4328 

and  : 

D  =  0.846 

This  is  very  nearly  lOi  ins.,  and  a  10  in.  pipe  would  answer, 
though  12  ins.  would  be  better. 

The  upper  section  must  deliver  2.6  cu.  ft.  per  second,  under 
a  grade  of  lTr50-y-.  Taking  the  same  probable  value  of  (7,  we  have  : 

6.25  X  0.66 

^  D*  = 

3.05 

whence  : 

D  =  1.237 

which  we  can  take  as  either  15  or  16  ins. 

This  diameter  might  have  been  obtained  from  that  of  the- 
lower  section,  by  means  of  (12).  Thus  : 

10      6.25 

D'  *  =   0.4328  X  -  X  — 
5         1 
D'  =  1.287 

This  last  formula  might  have  been  used  throughout,  but  (4) 
is  about  as  short  and  convenient ;  frequently  more  so. 

The  diameters  being  thus  determined,  the  quantities  should 
be  verified  by  (3).  They  will  be  found  somewhat  in  excess  of 
those  proposed,  owing  to  the  general  increase  of  the  diameters. 

As  regards  the  height  to  which  the  water  must  be  raised,  the- 
data  show  that  3  cu.  ft.  per  second  must  be  raised  to  a  sufficient 
height  to  reach  D  D'  at  an  elevation  of  (201)  above  datum.  lft\v& 
adopt  a  grade  of  YoVo»  ^ne  proper  diameter  of  the  pipe  would  be  : 

9  X  0.65 

z>*  =  — 

2.44 
D  =  1.32 

or, 

D  =  16  ins. 

If,  instead  of  pumping,  the  water  were  collected  in  a  reservoir 
by  damming  up  the  natural  flow  of  some  stream,  and  the  dam 
were  of  necessity  situated  at  an  elevation  so  great  that  a  clanger- 


46  PRACTICAL   HYDRAULIC    FORMULAE. 

ous  pressure  is  apprehended,  it  would  be  necessary  to  first  receive 
the  water  into  a  distributing  reservoir  situated  at  a  lower  level,  or 
else,  as  a  less  advantageous  expedient,  to  reduce  the  pressure  by 
gates,  properly  located  for  the  purpose. 

It  should  be  well  understood  that  all  the  above  assumed  data, 
particularly  such  as  relate  to  heads,  are  subjected  to  considerable 
variation  in  actual  practice.  All  the  calculations  have  been  based, 
of  necessity,  upon  the  hypothesis  that  the  exact  allotted  volume 
per  second  is  being  simultaneously  drawn  from  the  whole  system, 
This  would  rarely  be  the  case  ;  for  at  any  given  second,  the 
draught  would  be  liable  to  fluctuate  greatly  from  the  average. 
Indeed,  these  calculations  should  only  be  regarded  as  fixing,  with 
some  degree  of  approximation,  the  proper  relative  discharges  and 
pressures  at  the  different  points  supplied. 

The  remaining  north  and  south  pipes  should  be  calculated 
in  the  same  way.  Thus,  those  below  F  F'  on  the  west  side  dis- 
charge 1-6  cubic  ft.  with  a  grade  of  j^Vo"-  This  would  re- 
quire a  4  in.  pipe.  The  draught  from  these  would  somewhat 
lowerthe  piezometric  heads  at  their  junctions  with  the  side  mains. 
In  a  fine  calculation,  these  reductions  should  be  worked  out,  as 
was  done  in  the  previous  example  of  branch  pipes  ;  in  general, 
however,  and  in  cases  where  the  whole  supply  is  supposed  to  be 
carried  through  to  the  extremity  of  the  mains,  and  delivered  at 
the  highest  elevation,  as  was  done  in  the  present  instance,  and 
where  a  liberal  interpretation  has  been  given  to  the  calculation  of 
diameters,  this  is  not  indispensable.  At  the  same  time,  it  should 
be  a  guiding  principle  of  water-works  engineering  that  a  few 
hours  spent  in  the  office,  in  what  may  sometimes  be  considered 
an  over-refinement  of  calculation,  is  by  no  means  a  waste  of 
time,  and  frequently  enables  one  to  make  advantageous  and 
economical  modifications  in  a  project  of  distribution. 

It  may  here  be  noted  that  (12)  admits  of  being  put  into  a 
very  convenient  form  for  rapid  approximations.  To  do  this,  we 


PRACTICAL   HYDKAULIC   FORMULAE.  47 

luive  only  to  calculate  the  discharge  of  a  pipe  1  ft.  in  diameter, 
with  a  fall  of  1  ft.  per  thousand,  and  to  refer  all  other  discharges 
with  the  fall  per  thousand  feet  to  it,  in  order  to  obtain  the  cor- 
responding diameter.  The  quantity  discharged  by  the  above 
pipe  is  0.961  cu.  ft.  per  second,  and  the  square  of  the  same  is 
0.9^4.  Equation  (12)  may  then  be  written  : 


D=  A/  —  Xl.08 

or  very  nearly  : 

D=4/~  (13) 

we  have  also  very  nearly  : 


Q=   i'D*xH 

which  may  be  more  conveniently  expressed  thus  : 


Q  =  D-   V  D  X  H  (14  bis) 

We  have,  also, 


r  =   M  D  X  Hx  1.6  (Uter.) 

in  which  V—  velocity  in  feet  per  second. 

These  last  formulae,  it  will  be  perceived,  are  based  on  the  fact 
that,  given  a  certain  probable  degree  of  roughness,  a  pipe  1  ft.  in 
diameter,  with  a  fall  of  1  ft.  in  a  thousand,  will  deliver  1  cu.  ft.  of 
water  per  second.  If  we  desire  to  apply  them  to  smooth,  clean 
pipes,  we  have  only  to  Jialve  the  coefficient  for  a  12-in.  pipe,  which 
will  be  equivalent  to  writing  the  above  formula;  thus  : 


-  (15) 

•2H 


0-  =    I/7?8  X  2  H  (16) 

These  formulae  will  be  found  of  very  great  utility  in  arriving 
quickly  at  approximate  results.  They  can  be  advantageously  used 
in  sketching  out  a  network  of  pipes  such  as  we  have  just  been  con- 
sidering. To  facilitate  their  use  the  following  table  of  fifth  powers 


48 


PRACTICAL  HYDRAULIC   FORMULA. 


has  been  calculated.  This  table  indicates,  by  inspection,  the  diam- 
ters  in  inches  corresponding  to  the  fifth  roots  of  the  right-hand 
side  of  the  equations,,  expressed  in  feet. 


Diameters  in  incfies. 

Fifth  Powers  in  feet. 

Diameters  in  inches. 

Fifth  Poivers  infect. 

3 

0.000977 

22 

20.72 

4 

0  004115 

24 

32.00 

5 

0  01-256 

28 

47.75 

6 

0.03125 

28 

69.17 

8 

o.ru? 

30 

97.66 

10 

0.4019 

32 

134  9 

12 

l.OOOJ 

34 

182.6 

14 

2.1615 

36 

243.0 

16 

4.214 

40 

411.5 

18 

7  594 

42 

525.2 

20 

12  86 

48 

1,021  0 

All  the  diameters  which  have  been  already  calculated  can  be 
obtained  very  nearly  by  the  use  of  (13).  Relations  (13)  and  (14) 
might  also  have  been  used  in  some  of  the  previous  examples. 

Formulae  (13)  and  (14)  serve  to  show  the  comparatively  small 
influence  of  grade  as  affecting  the  volumes  discharged,  which 
point  has  been  already  alluded  to,  and  the  preponderating  influence 
of  diameter.  Thus,  we  see  by  the  above  formulae,  that  for  a 
diameter  of  1  ft.  and  a  fall  of  j^-,  the  volume  of  discharge  is  1 
cu.  ft.  If  we  wish  to  double  this  discharge  by  increasing  the  fall, 
we  must  adopt  a'grade  of  i^Vo?  *•  e->  we  must  quadruple  the  fall. 
If,  on  the  other  hand,  we  wish  to  produce  the  same  result  by  in- 
creasing the  diameter  without  changing  the  grade,  we  need  only 
adopt  a  diameter  of  1.32  ft.  and  even  a  little  less,  on  account  of 
the  decrease  in  the  coefficient.  That  is  to  say,  to  double  the 
discharge,  we  must  increase  the  fall  300  percent.,  or  the  diameter 
32  per  cent. 


NOTE.-  In  completion  of  what  has  been  already  said  in  this  chapter  (page  41 ).  re- 
garding the  limit  of  veloci'iea  for  pipes  of  different  diameters,  the  follow!ng  table 
(founded  upon  that  given  by  Mr.  Fanning)  indicates  pretty  closely  the  maximum 
/velocities  which  it  is  generally  advisable  to  produce  : 


Diameter  in  inches 

Velocity  in  ft.  per  sec 


2.5 


12 
3.5 


18 
4.5 


24 
5.5 


30 
6.5 


36 
7.5 


42 

8.5 


9.5 


CHAPTER  IV. 

Use  of  formula  (14)  illustrated  by  numerical  example  of  compound  system  combined 
with  branches—  Comparison  of  results—  Rough  and  smooth  pipes—  Pipes  com- 
municating with  three  reservoirs—  Numerical  examples  under  varying  condi- 
tions— Loss  of  head  from  other  causes  than  friction  -Velocity,  entrance  and 
exit  heads—  Numerical  example*  and  general  formulce—  Downward  discharge 
through  a  vertical  pipe—  Other  minor  losses  of  head—  Abrupt  changes  of  diam- 
eter—Partially opened  valve—  Branches  and  bends  -Certrifugal  force—  Small 
importance  of  all  losKes  of  head  except  fractional  in  the  case  of  long  pipes—  All 
such  covered  by  "even  inches"  in  the  diameter. 

As  an  illustration  of  the  use  of  (14)  we  will  calculate  by  its  aid 
the  discharge  from  a  reservoir,  tapped  at  a  depth  of  50  ft.  by  a 
horizontal  compound  system  consisting  successively  of  2,000  ft. 
of  12-in.  pipe,  2,000  ft.  of  24-in.  pipe  and  2.000  ft.  of  12-iiu 
Each  of  these  three  lengths  of  pipe  is  tapped  midway  by  a 
6-in.  pipe,  laid  horizontally,  'the  one  nearest  the  reservoir 
having  a  length  of  3,000  ft.;  the  next  1,000  ft.,  and  the  last  500 
ft.  (See  Fig.  9,  bis.)  All  the  pipes  being  open,  it  is  desired  to 
find  the  piezometric  heads  //,  //',  //",  h"  ',  k"",  at  each  branch  and 
change  of  diameter,  and  the  volumes  discharged  by  each  branch 
and  section  of  main  pipe. 

Beginning  at  the  lower  end  and  assuming  6  ft.  as  an  ap- 
proximate value  of  //,  we  have  from  (14),  H  always  representing 
thejfall  per  1,000: 


V  6  -f 

h'  =  15.36 


4      36  =    \  W(h^'  -  15.30) 
h"  =  15  65 


/  l/>.65 

/    •     —  = 


.  _  _  _ 

V  '9.  36  +  I/    •     —  =    V  3-'  (h'  '  '-  15.65) 
'        32 


50 


PRACTICAL   HYDRAULIC   FORMULAE. 
h"'=  16.09 


V  14.08  =    \'  h"  "  —  16.09 
h' ' ' '  =  30.17 

/10  06          

V-14.08  +  1/  =    Ix  h'  ""-30.17 

r  Q5! 


32 

h' '  "  '  =  48.i 


FIG.  9&. 


Comparing  this  value  with  the  given  height  50,  we  may  in- 
crease all  the  preceding  values  of  h,  U',  etc.,  in  the  proportion  of 

50 

—  .     But  in  practice  we  would  not  wish  to  reckon  on  the  total 

48.82 

head,  and  it  would  be  preferable  therefore  to  let  the  values  stand 
as  they  are. 

We  will  now  calculate  the  quantities,  calling  those  discharged 
from  the  successive  sections  of  main  pipe,  beginning  at  the  lower 
end,  Q,  Q',  Q",  Q'",  Q"",  and  Q'"",  and  those  discharged  by  the 
branches,  beginning  also  at  the  lower  end,  7,  q  ',  q",  respectively, 
using  both  (3)  and  (14).  The  results  given  by  (14)  naturally-check 
exactly,  since  they  depend  directly  upon  the  method  used  in  de- 
termining 7i,  h'j  etc. 

By  (3)  By  (14) 

Q        =2.39  2.45 

q        ----    .56  .61 

Q  +q        =  2.95  3.06 


PRACTICAL    HYDRAULIC    FORMULAE.  51 

By  (3).  By  (14). 

Q'      -  2.96  3.06 

Q"    =  2.99  3.05 

q'       =    .65  .70 

Q"  -f  q'  =  3.64  3.75 

Q'"  =  3.68  3.75 

Q""  =3.63  3.75 

q"  =    .52  .56 

Q"»  +  q"     =4.15  4.31 

#'""  =  4.13  4.32 

The  above  example  was  very  favorable  to  the  use  of  (14),  be- 
cause of  the  lengths  assumed  for  the  different  pipes,  but  in  almost 
all  cases  it  will  greatly  reduce  the  volume  of  calculation,  and 
frequently  give  sufficiently  close  results.  Indeed,  as  all  these 
calculations  are  merely  approximations,  and  as  we  have  taken  our 
coefficients  pretty  high,  it  would  no  doubt  often  be  found,  couM 
the  actual  discharges  be  measured,  that  the  apparently  less  exact 
formula  gave  the  more  correct  results. 

In  all  the  previous  examples,  the  coefficients  for  rough  pipes 
have  been  used.  It  is  well  to  remember  that,  as  is  shown  by  (15) 
and  (16),  the  discharge  of  a  clean  pipe  of  given  diameter  is  about  41 
per  cent,  greater  than  that  of  a  rough  pipe  of  the  same  diameter  ; 
also  that  the  diameter  of  a  clean  pipe,  discharging  an  equal  volume 
with  a  rough  one,  will  be  about  88  per  cent,  of  the  latter.  Be- 
tween these  limits  of  smoothness  and  roughness  there  are,  of 
course,  an  indefinite  number  of  gradations. 

A  very  interesting  investigation  is  that  of  a  system  of  pipes 
communicating  with  two  reservoirs,  and  discharging  either  freely 
in  the  air,  or  into  a  third  reservoir  situated  at  a  lower  elevation  aa 
shown  in  Fig.  10. 

A 

B  (60) 


52  PRACTICAL    HYDRAULIC    FORMULA. 

Let  us  suppose  the  water  surfaces  in  A  and  B  to  be  respect- 
ively 100  and  80   ft.  above  the  water  surface  in  C,  and  that  ulf 
the  pipes  shown  in  the  figure  are  12  ins.  in  diameter.     Let  the- 
total  length  of  pipe  from  A  to  C  be  4,000  ft. 

If  communication  were  shut  off  from  B,  the  flow  would  be- 
direct  from  A  to  C;  if  communication  were  shut  off  from  (7,  it- 
would  be  direct  from  A  to  />.  If  A  were  shut  off,  the  flow 
would  be  from  B  to  C.  If  all  the  communications  were  wide 
open,  we  desire  to  know  whether  the  flow  would  be  from  A  to 
B  and  C,  or  from  A  and  B  to  C1;  and  in  either  case,  to  know  the- 
piezometric  head  li,  at  the  junction  D,  and  the  volumes  dis- 
charged. 

First,  let  the  junction  D  be  situated  midway  in  the  4,000-ft. 
pipe  joining  A  and  C,  and  let  the  length  B  D  be  1,000  ft.  Let 
us  for  a  moment  revert  to  the  supposition  that  Bis  shut  off.  The 
flow  woul'd  then  be  from  A.  to  C,  the  hydraulic  grade  line  would 
be  a  straight  line  joining  the  surfaces  A  and  C,  and  under  our 
present  hypothesis,  that  the  junction  Dis  in  the  middle  of  A  C, 
the  piezometric  head  h  would  be  50  ft.  above  the  surface  of  the 
lower  reservoir  C.  But  B  is  supposed  to  be  80  ft.  above  the 
same,  and  therefore  the  flow  must  be  from  A  and  B  to  C.  We 
might  at  first  sight  suppose  that  the  flow  from  B  to  C  would  be 
in  virtue  of  the  head  80  —  50  —  30  ft.,  which  is  the  difference 
of  level  between  B  and  the  piezometric  head  at  the  junction; 
but  just  as  a  branch  drawing  water  from  a  main  pipe  lowers  the 
piezometric  head  at  the  junction,  so  does  a  branch  discharging 
into  the  main  pipe  raise  it.  It  is  necessary  to  see  what  the  height 
h  will  be  in  the  present  case. 

The  quantity  discharged  into  C  is  equal  to  the  sum  of  the 
quantities  passing  from  A  and  B.  All  areas  and  coefficients- 
being  equal,  and  all  reductions  made,  we  have  : 


1/7-  I/50-;4-  I/  8°- 


PRACTICAL   HYDRAULIC   FORMULAE.  53 

whence  : 


h  =  65+  y  4000  -  90  h  +  — 

and,  by  successive  approximations  : 

h  =  74 

Using  this  value  of  li  in  (3),,  we  obtain  the  different  dis- 
charges as  follows  : 

§=5.88 
=  3.18 
"  =  2.37 

This  gives  a  very  close  agreement  in  the  relation  Q  =  Q'  + 

«"• 

Suppose  now  that  the  diameter  of  the  branch  5  D  be  reduced 
to  6  in.,  all  the  other  conditions  remaining  the  same.  Still  regard- 
ing the  coefficients  as  equal,  in  order  to  get  rapidly  at  an  ap- 
proximation, factoring  the  areas  and  simplifying,  we  have  : 


whence  : 

16.5  h  =  840  +  4   VSOOO^ 

and,  by  successive  approximations  : 

h  =  58 

This  value  of  k  gives  the  following  quantities  : 

§=5.21 
=  4  43 
'  =  1.08 

A   tolerably  close   check,  but   showing  that  the  true  value   of  h 
is  a  little  greater  than  the  even  58  ft.  at  which  we  have  placed  it- 
Let  us  now  suppose  that  the  pipe  B  D  is  increased  to  a  diam- 
eter of  30  in.,  all  the  other  conditions  remaining  as  before. 

Then:  

\/?  =    4/^13+9    A/sTT^ 

'    2  2  V 

whence : 

h  =  79.90 

Giving  : 

=  6.111 
=  3.065 

}"  =  2.816 


54  PRACTICAL   HYDRAULIC    FORMULA. 

a  close  approximation  ;  the  true  value  of  h  lies  between  79.85  and 
79.90. 

As  li  increases  with  the  diameter  of  the  pipe  B  D,  it  might 
at  first  seem  as  though,  by  indefinitely  increasing  the  diameter,  li 
might  be  so  increased  as  to  cause  a  flow  from  A  into  B.  A  mo- 
ment's reflection,  however,  will  show  that  under  the  assumed  con- 
ditions the  diameter  can  never  be  sufficiently  increased  to  cause  a 
flow  toward  B.  For  it  has  been  seen  that  when  B  is  shut  off,  the 
piezometric  head  at  D  is  50  ft.  It  is  raised  by  opening  the  com- 
munication with  B,  and  allowing  water  to  flow  into  the  main  from 
B.  It  is  evidently,  therefore,  an  essential  condition  of  the  increase 
of  piezometric  height  that  the  flow  should  be  from,  not  to,  the 
reservoir  B. 

But  the  effect  will  be  different  if  the  junction/)  be  sufficiently 
advanced  toward  the  reservoir  A.  Let  us  suppose  the  positions 
of  the  three  reservoirs  to  remain  the  same,  all  the  pipe  diameters 
to  be  12  ins.,  and  the  point  of  junction  of  the  pipe  B  D  to  be 
placed  at  500  ft.  from  A  (Fig.  11).  If  communication  with// 
were  shut  off,  the  piezometric  height  at  D  would  be  87.5  ft. 
There  would  therefore  be  a  flow  from  A  to  B  and  C  when. the  pipe 
leading  to  B  was  open.  But  this  flow  would  not  take  place  under 
the  head  87.5,  for  the  draft  toward  B  would  lower  it. 


>?soo 

FIG.  11. 


To  ascertain  the  true  value  of  li  at  the  point  />,  we  have  the- 
relation  : 


500  3500  2500 


PRACTICAL   HYDRAULIC   FORMULAE.  55 

simplifying  ; 


/  /  h  /  h  -  80 

100  -  h  =  y  -  +  y 


47  h  =  4960  —  11.88  V  h*  -  80  h 

whence,  by  successive  approximations  : 

h  =  82.65 

Using  this  value  of  h  we  get : 

Q  =  5.695 
Q'  =  4.698 
Q"  =  0.995 

When  B  is  shut  off,  in  the  above  system,  the  discharge  from 
A  to  C  is  4.33  cu.  ft.  per  second. 

In  all  that  precedes,  only  the  resistance  due  to  friction  has 
been  considered,  and  the  total  difference  of  level  between  the 
source  of  supply  and  the  discharge  has  been  taken  as  available  for 
overcoming  this  frictional  resistance.  In  the  case  of  long  pipes, 
where  the  velocity  is  comparatively  low,  this  resistance  is  so  greatly 
in  excess  of  all  the  others  that,  in  order  to  simplify  calculations, 
they  are  neglected.  This  leads  to  no  material  error  in  cases  where 
the  pipe  is  over  1,000  diameters  in  length. 

Attention,  however,  has  been  already  called  to  the  fact  that 
there  are  other  resistances  which  require  a  certain  proportion  of 
the  total  head  to  overcome  them,  leaving  only  the  remainder 
available  as  against  friction.  Indeed,  it  is  evident,  if  we  assume 
all  the  head  to  be  consumed  by  frictional  resistance  alone,  the 
water  in  the  pipe  would  be  in  exact  equilibrium,  and  no  flow 
could  take  place. 

It  will  now  be  proper  to  show  how  the  total  loss  of  head, 
from  all  causes,  may  be  calculated.  And  first,  a  word  in  refer- 
ence to  the  phrase  "  loss  of  head ''just  employed.  This  term, 
often  met  with  in  treatises  on  hydraulics,  may  occasionally  prove 
confusing.  It  is  really  little  more  than  a  convenient  abbrevia- 
tion. When  we  speak,  for  instance,  of  "  the  loss  of  head  due  to 
velocity,"  we  mean  the  head,  or  fall,  theoretically  necessary  to 


56  PRACTICAL   HYDRAULIC   FORMULA. 

produce  that  velocity.  Similarly,  when  we  speak  of  "  the  loss  of 
head  due  to  resistance  to  entry,"  we  mean  the  amount  of  head, 
or  pressure,  necessary  to  force  the  fluid  vein  into  the  mouth  of  the 
pipe  or  orifice,  against  the  resistance  of  its  edges.  This  resist- 
ance, it  may  be  remarked  in  passing,  as  well  as  that  due  to  bends, 
elbows  and  branches,  shortly  to  be  mentioned,  is  caused  by  the 
fact  that  water  is  not  a  perfect  fluid,  and  therefore  changes  of  di- 
rection in  its  flow  require  a  certain  amount  of  force  to  break  or 
distort  the  form  of  the  fluid  vein  as,  though  to  a  very  much  less 
degree,  would  be  the  case  with  a  plastic  body  under  similar  cir- 
cumstances. The  property  of  water  which  causes  these  resist- 
ances is  called  its  viscosity. 

As  applied  to  long  pipes,  the  principal  "  loss  of  head,"  and 
fclie  only  one  hitherto  considered,  is  the  f fictional.  The  term 
thus  applied  means  the  height  or  pressure  necessary  to  overcome 
the  friction  of  the  water  passing  with  a  given  velocity  through  a 
pipe  of  given  length  and  diameter.  Thus,  when  we  speak  of  the 
frictional  loss  of  head  per  1,000  ft.  in  reference  to  a  given  pipe, 
we  mean  the  fall  per  1,000  ft.  necessary  to  maintain  the  given  or 
desired  velocity,  as  against  friction. 

We  will  now  investigate  this  subject  by  means  of  the  follow- 
ing problem  :  Two  reservoirs  (Fig.  12)  containing  still  water 
and  having  a  difference  of  level  of  30  ft.,  are  joined  by  a  pipe  12 
ins.  in  diameter  and  3,000  ft.  long.  What  is  the  velocity  of  dis- 
charge between  the  upper  and  lower  reservoirs  ? 


FIG.  12. 

From  what  has  been  already  said,  it  will  be  seen  that,  besides 
the  frictional  loss  of  head,  there  will  be  the  loss  of  head  due  to 


PRACTICAL   HYDRAULIC    FORMULAE.  57 

velocity  and  that  due  to  entrance.  If  the  pipe  discharged  freelv 
in  the  air  at  its  lower  end,  at  the  vertical  distance  of  30  ft.  below 
the  surface  of  the  water  in  the  upper  reservoir,  these  three  would 
be  the  only  losses  of  head  incurred,  and  their  sum  would  be  equal 
to  30  ft.;  but  as  the  discharge  takes  place  in  a  reservoir,  the  sur- 
face of  the  water  in  which  is  supposed  to  cover  the  end  of  the 
pipe,  to  a  sufficient  depth  to  cause  the  discharge  to  take  place  in 
still  water,  there  is  the  further  loss  of  head  due  to  the  extinction 
of  the  velocity  which  is  dissipated  in  vortices.  This  loss  consti- 
tutes what  may  be  called  the  back  pressure  of  the  reservoir. 

In  solving  this  problem,  let  us  first,  as  heretofore,  neglect 
all  losses  except  frictional  ones.  AVre  have  then,  from  (1),  using 
the  above  data,  and  the  coefficient  for  rough  pipes  : 

i 

=    0.00066  V* 

100 
•       V*  =  15  15 

V  =   3.89ft.  per  second. 

The  head  theoretically  necessary  to  produce  this  velocity  is 
given  by  the  formula  derived  from  the  law  of  falling  bodies, 

F2 

h  3  -^-  by  substitution  of  the  above  value  F.     Thus  : 
*ff 

15.15 

~&rr 

h=   0.2352 

Besides  this,  there  is  the  loss  of  head  due  to  entrance.  We 
have  already  seen  that  this  is  always  equal  to  about  half  the 
velocity  head.  We  have  then  : 

h 

h  4-  —  =  0.3528 
2 

The  loss  of  head  from  back  pressure  of  the  water  in  the  lower 
reservoir,  being  that  necessary  to  extinguish  the  velocity,  must  be 
-equal  to  that  necessary  tb'produce  the  same.  We  have,  therefore, 
for  the  total  losses,  outside  of  friction  : 


58  PRACTICAL   HYDRAULIC    FORMULA. 

And  the  head  available  for  overcoming  friction  becomes 

30  —  0.588  =  29.412 

We  must  now  recast  our  original  calculation,  using  29.4  ft. 
instead  of  30  as  available  frictionai  head.  Thus  : 

29.4 

=    0.00066  F2 

3000 

F2  =  14.3 
V   =    3.35 

This  is  a  very  small  reduction  from  the  velocity  already  ob- 
tained. But,  in  order  to  see  how  our  previous  solution  is  affected 
by  the  change,  we  will  work  out  new  values  for  the  subheads. 

_  14.8 

~GIA 

h  =  0.23 

h 

h  -f  —  +  h  =  0.575 
2 

30-0.575  =  29.425, 

leaving  the  previous  valu^  practically  unchanged. 

Let  us  now  see,  by  means  of  a  general  formula,  what  is  the 
amount  of  error  which  we  commit  when  we  ignore  all  resistances 
except  friction. 

Calling  Fthe  actual  mean  velocity,  that  is  the  actual  volume 
discharged  divided  by  the  area  of  the  pipe  (3),  we  have,  in  the  case 
of  discharge  between  two  reservoirs,  as  shown  in  Fig.  12,  the 
following  subheads,  which  together  make  up  the  total  head  //: 

F2       F2       F2      L  C  Fa 
~  20       40       20  D 

5  F2      L  C  F2 

H  = -f- 

40  D 

JJ=0.039F2-f  Z.CF3 
D 

That  is  to  say,  by  using  (3),  which  gives 

H  =  L  C  F2 


PRACTICAL    HYDRAULIC    FORMULA.  59 

we  make  the  error  of  omitting  a  height  not  quite  equal  to  4  per 
cent,  of  the  square  of  the  velocity. 

In  long  pipes  this  is  a  very  trifling  amount. 

If  the  pipe  discharged  in  free  air,  we  would  have  : 
v*     v*     L  c  v* 

H=  —  4  —  +  — 
20       ig  D 

H  =  0.0233  F2  +  L  C  V* 


Iii  this  case  we  make  the  still  smaller  error  of  omitting  %\% 
of  F»- 

In  all  cases,  having  obtained  V9  by  means  of  (1),  we  can 
easily  judge  from  the  nature  of  the  problem  whether  it  is  neces- 
sary to  take  account  of  these  errors.  In  designing  a  system  of 
pipes,  where  the  problem  generally  is  to  find  the  proper  diameter 
for  a  certain  discharge,  the  practice  of  taking  the  next  highest 
even  inch  will  almost  always  amply  suffice  to  cover  all  omissions. 

As  has  been  already  stated,  in  all  ordinary  circumstances  of 
pipelaying,  the  horizontal  measurement  of  the  pipe  is  taken  in- 
stead of  its  actual  length.  It  is  only  in  special  cases  that  this 
cannot  be  done.  The  extreme  limit  occurs  in  the  case  of  a  ver- 
tical pipe  discharging  from  the  bottom  of  a  reservoir.  This  con- 
stitutes a  very  interesting  special  case,  for  should  the  reservoir  be 
of  indefinitely  large  area,  but  of  relatively  shallow  depth,  the  rela- 

H 
tion  —  tends  toward  unity  as  //,  and  consequently   H,  increase. 

Li 
The  velocity,  as  determined  by  (1),  tends  therefore  toward  : 


and  remains  constant,  no  matter  how  greatly  L  may  be  increased. 
If  we  apply  this  formula  to  a  12-in.  pipe  of  indefinite  length, 
using  the  coefficient  for  rough  pipes,  we  get, 

V=  38.9 


60  PRACTICAL  HYDRAULIC   FORMULAE. 

This  is  the  maximum  velocity  of  discharge  in  feet  per  second 
for  a  vertical  12-in.  pipe  under  the  given  circumstances.* 

There  are  several  minor  losses  of  head,  besides  those  already 
considered,  which  are  liable  to  occur  from  changes  of  diameter, 
branches,  and  bends  or  elbows.  Our  experimental  knowledge  of 
the  effects  of  these  features  is  very  limited,  and  it  is  probable 
that  much  weight  should  not  be  attached  to  the  formulae  given 
for  their  determination.  A  brief  space  will  be  devoted  to  their 
consideration,  more  with  a  view  to  make  the  present  paper  com- 
plete than  for  any  practical  value  which  they  possess. 

When  water  passes  through  a  pipe  of  which  the  diameter  is 
abruptly  Changed,  at  a  certain  point,  to  a  greater  or  a  smaller  one, 
there  is  a  loss  of  head  due  to  the  eddies  formed  and  the  sudden 
contraction  of  the  fluid  vein.  In  practice  such  pipes  are  always 
joined  by  a  reducer,  or  special  casting,  which  forms  a  tapering 
connection  between  the  two.  This  greatly  diminishes  the  agita- 
tion of  the  water  in  passing  from  one  pipe  to  the  other.  It  would 
seem,  however,  that  the  mere  change  of  velocity,  independent  of 
such  agitation,  causes  some  slight  modification  of  the  profile  of 
the  hydraulic  grade  line  ;  and  it  will  be  well,  in  any  event,  to 
give  formulae  for  the  different  cases  which  may  occur  when  abrupt 
changes  take  place,  as  these  give  rise  to  the  maximum  retardation. 
The  following  formulae  are  taken  from  Claudel's  Aide  Memoir  e, 
ninth  edition. 

First. — When  the  change  is  from  one  pipe  to  another  of 
smaller  diameter,  we  have  : 

whence:  *  =  0'49?7 

h  =  0.00076  V2 

V  being  the  velocity  of  the  water  in  the  smaller  pipe.  We  have 
seen,  by  examples  previously  given,  how  this  velocity  may  be 
obtained. 

*  The  same  result  may  be  inferred  from  what  has  been  said  in  Chapter  I.  about  a 
pipe  la;d  so  lhat  its  axis  coincides  with  the  hydraulic  parade  line.  Obviously,  a 
vertical  pipe  discharging  downward  is  a  special  case  of  such  coincidence. 


PRACTICAL   HYDRAULIC    FORMULA.  61 

Second. — If  the  water  (Fig.  13),  in  its  passage  from  the  greater 
to  the  smaller  pipe,  passes  through  an  opening  in  a  thin  diaphra 


FIG.  is. 
as  in  the  case  of  a  partially  opened  stop-cock,  we  have  : 


2g   \0.62S' 

in  which  Fis  the  velocity  in  B,  £the  area  of  cross-section  of  Bf 
and  S'  the  area  of  the  opening  in  the  diaphragm. 

Third.  —  When  the  flow  is  from  one  pipe  to  another  of  larger 
diameter  : 


in  which  F  =•  velocity  in  small  pipe,  and  V  —  velocity  in  larger 
one.  When  the  water  passes  from  a  pipe  into  a  reservoir,  as  in 
the  case  lately  considered,  V  becomes  zero,  and  we  have,  as  already 
established  in  that  case  : 

ra 
h  =  — 

20 

Another  loss  of  head  is  that  due  to  branches  (Fig.  14).     In. 


c 
FIG.  14. 


this  case  the  water  flowing  from  A,  with  a  velocity  F,  is  split  at 
the  junction,  part  passing  on  toward  B,  with  a  reduced  velocity 
F',  and  part  entering  the  branch  and  flowing  toward  (7,  with  the 
velocity  V".  The  loss  of  head  occasioned  by  perturbations  of  the- 
water  at  the  junction  has  not  been  satisfactorily  investigated. 


(52  PRACTICAL    HYDRAULIC    FORMULA. 

When  the  branch  leaves   the  main  at  a  right  angle,  this  loss,  as 
determined  by  a  few  incomplete  experiments,  is  : 

3  v* 

~    20 

V"  being  the  velocity  in  the  branch.     We  have  already  seen  how 
this  velocity  may  be  calculated. 

If,  as  is  generally  the  case  in  practice,  the  branch  is  deflected 
gradually  instead  of  forming  an  abrupt  angle  of  90°,  the. vortices 
are  nearly  annulled,  and  the  only  loss  can  be  from  the  difference 
of  the  velocities  in  the  three  pipes.  Thus  for  B  and  C,  respec- 
tively, we  have  : 

/v-  v  * 
h  =  ( 

\     20     ) 
(V 
h'  = 


For  bends,  or  elbows,  Navier's  formula  for  loss  of  head  is  : 
F2  /  \  A 

h  =  —  I  0.0123  +  0.0183  R  1  - 
20  \  '  R 

in  which  V  =  velocity  of  flow,  72  =  the  radius  of  the  bend,  taken 
along  the  axis  of  the  pipe,  and  A  =the  length  of  the  bend,  also 
measured  along  the  axis. 

It  will  readily  be  seen  how  very  trifling  the  loss  of  head  from 
this  cause  will  be  in  all  ordinary  cases. 

The  water  passing  around  a  bend  exercises  a  radial  thrust 
upon  it  which  may  sometimes  be  so  considerable  as  to  require 
bracing  against.  The  expression  for  the  centrifugal  force  Fis  : 

M  V* 

F= 

R 

in  which  J/=the  mass  of  the  liquid  in  motion,  V=  its  velocity, 
and  R=  the  radius  of  the  bend  measured  on  its  axis. 

As  an  illustration,  we  will  suppose  a  pipe  24  ins.  in  diameter, 
through  which  the  water  flows  with  the  velocity  of  8  ft.  per  sec- 
ond, around  a  bend  of  8  ft.  radius. 


PRACTICAL   HYDRAULIC   FORMULA.  tf3 

The  mass  of  the  liquid  in  motion  is  its  weight  divided  by  g. 
The  centrifugal  force,  therefore,  per  running  foot  is  : 

3.14  v  62.5       82 

32.2  8 

F  =  48.721bs. 

If  the  bend  turns  a  quarter  circumference,  its  development 
on  the  axis  will  be  12.57  ft.,  and  the  total  thrust  on  the  bend  will 
be  48.72  x  12.57  =  612.4  Ibs. 

This  would  be  liable  to  be  intensified  by  sudden  changes  in 
velocity,  and  if  the  bend  is  not  well  abutted,  might  tend  to  draw 
the  joints. 


FIG.  15. 

Fig.  15  shows  the  manner  in  which  such  losses  of  head  as  we 
have  been  just  considering  modify  the  the  profile  of  the  hydraulic 
grade  line.  The  dotted  line  shows  the  grade  as  determined  by 
the  calculations  which  we  have  already  made  for  a  line  of  pipes 
of  varying  diameter.  The  full  line,  broken  at  the  reservoir  and 
at  each  change  of  diameter,  shows  the  hydraulic  grade  as  modified 
by  losses  of  head  due  to  velocity  and  changes  of  diameter.  It  will 
be  understood,  of  course,  that  this  is  a  mere  random  sketch,  with- 
out reference  to  proportion. 

The  result  of  what  precedes  in  reference  to  all  losses  of  head 
other  than  friction  shows  that  in  practice,  and  in  the  case  of  long 
pipes,  such  losses  exercise  but  a  trifling  influence.  A  very  small 
increase  in  the  diameter  of  the  pipe  over  that  obtained  by  calcula- 
tion based  on  fractional  head  alone,  such  as  would  naturally  be 
made  to  get  even  inches,  will  in  almost  all  cases  largely  cover  all 
losses  due  to  velocity,  entrance,  branches,  bends,  etc. 


,  CHAPTER   V. 

XOTES   01*   PlPELAYING. 

It  will  not  be  amiss  at  present  to  give  some  hints  respecting 
Pipelaying.  Enough  has  been  already  said  to  show  how 
greatly  the  smoothness  or  roughness  of  the  interior  of  a  pipe 
affects  the  velocity  of  the  flow  of  water  through  it.  A  line  of 
pipes  is  made  up  of  a  great  number  of  separate  lengths  joined 
together,  generally  by  the  spigot  end  of  each  pipe  entering  into 
the  hub  or  bell  end  of  the  other.  Each  of  these  joints  occasions 
more  or  less  friction,  and  it  is  essential,  not  only  on  this  account, 
but  also  and  more  particularly  in  order  to  make  a  substantial  and 
enduring  piece  of  work,  that  the  pipes  should  be  laid  as  evenly  as 
possible,  and  the  joints  well  fitted  and  calked.  The  alinement 
should  be  straight  and  the  grade  regular.  This  latter  is  the  more 
important  of  the  two,  because  sags  and  depressions  in  the  line 
occasion  deposits  of  impurities  in  the  low  points  and  accumula- 
tions of  air. in  the  high  ones.  The  line  should  run  straight  and 
even  between  changes  of  grade  and  direction.  Each  low  point, 
or  point  from  which  the  grade  rises  both  ways,  should  be  pro- 
vided with  a  special  •'  blow-off "  and  stop  cock,  to  clear  it  of 
sediment  by  blowing  off  under  pressure,  and  each  summit,  or 
point  from  which  the  grade  falls  both  ways,  should  have  a  special 
air  vent,  or  hydrant,  to  discharge  the  accumulated  air  from  time 
to  time.  When  a  new  line  of  main  is  filled  for  the  first  time,  or 
When  a  line  is  refilled  after  having  been  emptied  for  any  cause, 
all  the  blow-offs  and  air  cocks  should  be  opened,  and  the  water 


PRACTICAL   HYDRAULIC   FORMULAE.  65 

admitted  very  slowly  by  giving  a  few  turns  only  to  the  admission 
valve.  Then,  as  the  pipe  gradually  fills,  the  blow-offs  should  be 
closed  progressively  as  the  water  reaches  them,  and  the  air  cocks 
also,  beginning  to  close  the  latter,  if  possible,  from  the  lower  end, 
and  only  when  they  discharge  solid  water.  Changes  of  horizontal 
direction  should  be  joined  by  as  easy  curves  as  can  be  obtained. 
In  sharp  curves  and  large  diameters,  special  curved  pipe  may  be 
necessary,  but,  in  general,  curves  are  got  in  with  straight  pipe, 
using  all  short  pieces  that  may  be  on  hand,  and,  if  necessary,  cut- 
ting whole  pipe,  and  joining  the  straight  pieces  with  sleeves. 

In  a  well-laid  pipe  line,  all  pipe,  particularly  all  those  of  20" 
diameter  and  upward,  are  laid  on  blocks.  These  blocks  consist 
of  pieces  of  wood  sawed  out  to  regular  dimensions,  there  being 
two  under  each  pipe,  one  just  behind  the  hub  and  the  other  as 
near  the  spigot  end  as  will  permit  of  the  joint  being  easily  reached 
for  calking,  say  about  2  feet.  For  diameters  from  36"  to  48", 
the  length  of  these  blocks  may  be  equal  to  the  diameter  of  the 
pipe,  and  about  a  foot  wide  and  6"  thick.  For  smaller  pipe  they 
may  be  about  two  feet  longer  than  the  diameter  of  pipe  and  pro- 
portionately lighter  than  for  the  larger  sizes.  The  pipe  is  held 
in  its  place  on  these  blocks  by  means  of  wooden  wedges  placed 
side  by  side,  on  opposite  sides  of  the  pipe,  and  driven  past  each 
other.  For  48"  pipe  these  wedges  may  be  about  18"  long,  6"  wide 
and  4"  thick  at  the  butt.  For  smaller  pipe  they  will  of  course  be 
lighter. 

The  instrumental  alinement  of  the  pipe  line  presents  no 
particular  difficulty,  because  the  excavation  once  correctly  started 
is  not  likely  to  deviate  to  any  injurious  extent.  It  is  much  more 
difficult,  as  it  is  more  important,  to  keep  the  grade.  This  is  best 
effected,  practically,  I  think,  in  the  following  manner  :  Let  the 
ordinary  marked  grade  stakes  be  set  for  the  excavation.  Then 
when  the  proper  depth  has  been  reached,  or  nearly  so,  let  grade 
plugs  be  driven  in  the  bottom  of  the  trench,  every  50  feet  or 


66  PRACTICAL   HYDRAULIC    FORMULAE. 

oftener,  with  their  heads  exactly  to  grade.  .  A  line  can  then  be 
stretched  from  one  to  the  other,  and  the  blocks  laid  to  it.  It  is 
better  to  bed  the  blocks  a  trifle  low,  say  a  quarter  to  half  an  inch, 
particularly  with  heavy  pipe  and  hard  bottom,  and  then  raise 
the  pipe  to  grade  by  driving  in  the  wedges.  It  is  not  necessary 
to  set  the  pipes  with  rod  and  level.  If  the  grade  pings  have 
been  driven  as  suggested,  a  competent  foreman  will  adopt  any 
one  of  many  ways  for  sighting  in  the  pipe  to  the  proper  level. 
With  soft  ground  and  heavy  pipes,  longitudinal  stringers  are  ad- 
vantageously employed  under  the  blocks,  the  spaces  between  them 
being  well  packed  with  broken  stone  or  other  ballast. 

When  pipe  have  been  laid  and  calked,  it  is  advantageous  to 
cover  them  as  soon  as  possible  by  backfilling  the  trench,  to 
prevent  the  joints  from  drawing  in  consequence  of  expansion  and 
contraction  due  to  exposure  to  the  changes  of  temperature  of  the 
air. 

In  backfilling  the  trench  after  the  pipe  is  laid,  be  very  care- 
ful that  the  earth  is  well  tamped  in  under  the  pipe,  so  that  it  may 
have  a  solid  bearing  throughout  its  entire  length.  The  earth  put 
in  next  to  the  pipe  should  be  clean  and  free  from  stones.  Be 
particularly  careful  that  no  large  stone  gets  under  the  pipe,  as  in 
case  of  a  sudden  jar,  such  as  would  be  produced  by  a  casual 
"  water  hammer," -it  might  punch  a  piece  out  of  the  pipe,  or  at 
least  crack  it. 

In  leading  and  calking  joints,  the  specifications  generally 
call  for  a  certain  depth  of  lead.  The  specifications  of  the  city  of 
New  York  require  4  ins.  of  lead  for  48-in.  pipe.  It  is  a  great 
advantage  to  have  a  deep  joint,  although  the  necessity  is  some- 
times denied  upon  the  ground  that  in  calking  it  is  impossible  to 
"upset"  the  lead  to  a  depth  of  more  than  perhaps  half  or  fof 
an  inch,  and  that  therefore  the  additional  lead  is  of  no  advantage. 
This  is,  however,  a  mistake,  for  an  abundant  depth  must  be 
allowed  for  cutting  off  the  protruding  lead  in  recalking  a  joint 
which  has  drawn.  The  lead  which  is  drawn  out  when  a  line  of 


PRACTICAL  HYDRAULIC   FORMULA  67 

pipe  contracts  is  not  forced  back  again  when  the  pipe  expands  ; 
on  the  contrary  it  remains  out,  and,  if  the  pipe  contracts  again,  a 
fresh  amount  will  probably  be  again  drawn  out,  all  of  which 
must  be  cut  off  when  the  joint  is  recalked. 

In  calking  joints,  yarn  or  gasket  is  first  driven  in  until  the 
proper  depth  is  secured,  and  lead  is  then  poured  in,  either  by  the 
use  of  the  clay  dam  or  *'  snake,"  or  by  the  use  of  a  metallic 
•'"clip."  The  latter  is  very  useful  in  laying  large  pipe,  as  it  en- 
able the  whole  joint  to  be  run  at  one  pouring.  A  refinement  in 
running  joints  is  to  employ  a  lead  gasket,  for  example  a  piece  of 
lead  pipe,  hammered  flat,  and  circled  around  the  pipe.  It  is 
driven  in  and  calked,  and  the  remainder  of  the  joint  is  then  run 
and  calked  in  the  usual  manner.  The  advantages  of  this  system 
are  that  there  is  no  perishable  material  used,  and  that  the  joint  is 
calked  on  both  faces,  which  is  very  favorable  to  making  it  tight. 
It  is  sometimes  difficult  to  get  the  trench  and  pipe  sufficiently  dry 
to  admit  of  pouring  a  molten  joint.  In  such  cases  it  is  necessary 
to  put  the  lead  in  cold,  perhaps  in  the  form  of  a  ring  of  flattened 
lead  pipe,  as  above,  and  calk  as  put  in.  This  makes  a  very 
satisfactory  joint,  only  it  is  slower  and  more  expensive. 

Mr.  Billings,  in  his  excellent  treatise,  (e  Some  Details  of 
Water- Works  Construction,"  gives  Mr.  Dexter  Brackett's  formula 
for  the  average  weight  of  lead  in  a  joint  about  2|  in.  deep,  as 
follows  : 

l  =  2d, 

in  which  I  =  pounds  of  lead  per  joint,  and  d  =  inside  diameter 
of  pipe  in  inches.  As  the  pipes  are  usually  12  ft.  in  length,  the 
weight  of  lead  per  running  foot  equals  one-sixth  of  diameter  of 
pipe  in  inches.  If  a  4-in.  joint  is  used,  we  would  have 

I  =  3.2d, 

and  the  weight  per  running  foot  under  above  assumption  of  length 
d  d 

of  pipe  would  be  ,  or,  in  round  numbers,  — . 

3.75  4. 


APPENDIX. 

As  it  is  convenient  in  making  estimates  to  have  the  correct  weight  of 
cast  iron  pipes  of  different  diameters  in  handy  form,  I  subjoin  a  table  of 
weights  of  pipes  made  by  the  Warren  Foundry,  of  Phillipsburg,  N.  J.: 

TABLE  SHOWING  THICKNESS  OF  METAL  AND  WEIGHT  PER    LENGTH  FOR    DIFFERENT 
SIZES  OF  PIPE  UNDER  VARIOUS  HEADS  OF  WATER. 


g 

50  ft,  head. 

100  ft.  head. 

150  ft.  head. 

200  ft.  head, 

250  ft.  head. 

300  ft.  head. 

It 

IS 

I 

P 

L 

11 

1. 

ii 

1. 

•3 

1. 

§1 

| 

e 

SB 

|| 

if 

JS 

ft 

•sS 

|| 

••—  *M 

If 

ji 

If 

|1 

If 

c  QJ 

•33 

|| 

1 

2  ° 

£~ 

ir 

J2 

S° 

£~ 

H® 

£~ 

5*0 

g- 

g° 

£~ 

2 

.294 

63 

.312 

67^ 

.330 

72 

.348 

76Vfc 

.366 

81 

.384 

8ft- 

3 

.314 

144 

.353 

149 

.36* 

153 

.371 

157 

380 

161 

.390 

166 

4 

.361 

197 

.373 

204 

.385 

211 

.397 

218 

.409 

226 

.421 

235 

5 

.378 

254 

.393 

265 

.408 

275 

.423 

286 

.438 

298 

.453 

309 

6 

.393 

315 

.411 

330 

.429 

345 

.447 

361 

.465 

377  j 

.483 

393 

8 

.422 

445 

.451)!      475 

.474 

502 

.498 

529 

.522 

557 

.546 

584 

10 

.459 

eoo 

.489!      641 

.519 

682 

.549 

723 

.579 

766j 

.609 

808 

12 

.491 

768 

.527       826 

.563 

885 

.599 

944 

.635 

1,0  )4| 

.671 

1,064 

14 

.524 

954 

.566    1,031 

.608 

1,111 

.650 

1,191 

.692 

1,272; 

.734 

1,352 

16 

.580 

1,215 

.604    1,253 

.652 

1,360 

.700 

1.463 

.748 

1,568! 

.796 

1,673 

18 

.589 

1,370 

.643    1,500 

.697 

1,630 

.751 

1,761 

.805 

1,894; 

.859 

2.02ft; 

20 

.622 

1.603 

.6821   1,763 

.742 

1,924 

.802 

2,086 

.862 

2,->48| 

.922 

2.412 

24 

.687 

2.120 

.759 

2,349 

.831 

2,580 

.903 

2,811 

.975 

3,045! 

1.047 

3.279' 

3'J 

.785 

3,020 

.875 

3,376 

.965 

3,735 

1.055 

4,095 

1.145 

4.4581 

1.235 

4,822 

36 

.882 

4,070 

.990    4,581 

1.098 

5,096 

1.206 

5,613 

1.314!   6,133; 

1.422 

6,656 

42 

.989 

5,265 

1.106    5,9.i8 

1.232 

6657 

1.358 

7.360 

1.484!   8,070; 

1.610 

8,804 

48 

1.078 

6616 

1.222    7,521 

1.366 

8431 

1.510 

9,340 

1.654  10,269 

1.798 

All    pipe  cast  vertically  in  dry  sand,  in  lengths  of  12   ft.,  except  the  2-in., 
which  are  cast  9  ft.  long. 


The  general  formula  for  weight  of  cylindrical  cast  iron  pipe  of  givem 

thickness  is : 

W  =  0.82  (D  +  T)  T  X  L,  (1) 

in  which 

W  -  weight  in  pounds. 
D  =  inside  diameter  in  inches. 
T  =  thickness  of  metal  in  inches. 
L  =  length  iu  inches. 


PRACTICAL   HYDRAULIC   FORMULAE.  69 

A  convenient  approximate  formula  for  the  weight  per  foot  of  cylin- 
drical part  of  a  cast  iron  pipe  is  : 

W  =  10  (D  +  T)  T.  (2) 

In  calculating  the  weight  of  cast  iron  pipe,  they  are  always  considered 
as  being  cylindrical,  and  eight  inches  of  length  are  added  as  the  equiva- 
lent of  the  hub  or  bell  for  all  diameters.  Thus  a  pipe  measuring  12  feet 
•over  all,  including  the  hub,  would  be  considered,  as  regards  weight,  as  a 
plain  cylindrical  pipe  12  feet  8  inches  long.  For  instance,  to  calculate 
the  weight  of  a  42-inch  pipe,  12  feet  over  all  and  0.980  inch  thick,  by  the 
above  approximate  formula  (2)  we  have: 

W  —  10  X  12.67  X  42.98  X  0.98  =  £336.6  Ibs. 

This  is  about  \\%  in  excess  of  weight  given  in  Warren  Foundry 
iable. 

If  we  wish  to  find  the  thickness,  the  length,  diameter  and  weight  be- 
ing given,  we  have  from  (1): 


/   w       D*     n 

T=\/  -      -+    - 

V    Q.S2L  4          2 


To  find  the  proper  thickness  in  inches  corresponding  to  the  head  in 
feet,  H,  of  pressure  for  a  given  diameter  in  inches,  D,  we  have  : 

T  =  0.00006  H  D  -f  0.0133  D  +  0.296. 

The  hydraulic  engineer  will  find  it  both  interesting  and  useful  to 
work  out  concise  formulse  covering  frequently  recurring  cases,  and  enter 
them  in  his  notebook  for  future  reference.  These  should  be  carefully 
•checked  by  testing  them  numerical  I y,  so  that  they  can  be  used  with  con- 
fidence when  wanted.  For  instance,  it  is  a  common  practice  to  allow  100 
United  States  gallons  per  capita  to  be  consumed  in  10  hours,  in  calculat- 
ing the  proper  supply  for  a  town.  To  find  the  diameter  of  a  pipe  to  con- 
vey this  amount  we  may  use  the  approximate  formula  : 


H 
in  which 

D  =  diameter  in  feet. 

H=  fall  per  1,000. 

N '=  number  of  population  to  be  supplied. 

Also,  in  estimating  theoretical  horse  power  necessary  to  raise  a  given 
volume  of  water  to  a  certain  height  and  in  a  certain  time,  we  have 


70  PRACTICAL   HYDRAULIC    FORMULAE. 

and 

_      ' 


5.7 
in  which 

Q  =  cubic  feet  per  second  . 

Q'  =  millions  of  U.  S.  gallons  per  24  hours. 

H  =  lift  in  feet. 

In  calculating  the  power  necessary  to  pump  water  into  a  reservoir 
situated  at  certain  horizontal  and  vertical  distances  from  the  pumps,  and 
connected  with  them  by  a  force  main  of  given  diameter,  we  must 
imagine  the  water  to  be  raised  to  a  certain  elevation  above  the  reservoir 
such  that  the  difference  of  level  between  this  elevation  and  that  of  the 
reservoir  shall  be  sufficient  to  convey  the  required  amount  of  water  to  the 
reservoir  against  the  friction  of  the  force  main.  Thus,  suppose  we  wished 
to  deliver  1  cubic  foot  per  second  through  a  12  inch  force  main  to  a  reser- 
voir 100  feet  above  the  pumps,  and  distant  10,000  feet  from  them.  By  our 
approximate  formula  (13)  we  know  that  this  requires  a  fall  of  one  foot  in 
a  thousand,  so,  as  the  distance  is  10,000  feet,  we  need  10  feet  to  overcome 
the  friction,  and  our  pumps  must  therefore  be  able  to  raise  the  given 
volume  a  total  height  of  110  feet.  This  calls  for  12.47  theoretical  horse 
power,  by  the  formula  just  given. 


SECOND  PART. 


NOTES  ON  WATER  SUPPLY  ENGINEERING. 


QUALITY  OF   WATER. 

The  first  question  in  regard  to  a  water  supply  is,  evidently, 
quality.  The  solution  of.  this  question,  either  generally  or  in  any 
particular  case,  is  by  no  means  a. simple  nor  any  easy  one,  partic- 
ularly since  it  is  now  fully  recognized  that  chemical  analysis,  to 
which  we  naturally  recur  in  such  cases,  is  far  from  being  a  final 
criterion,  but  is  only  one  element  in  a  group  of  data  from  which 
we  infer  the  probable  quality  of  a  given  source  of  supply. 

As  animal  refuse  and  the  waste  products  of  human  -industry 
are  the  principal  sources  of  menace  to  a  water  supply,  we  com- 
monly look  for  a  high  degree  of  purity  in  water  drawn  from  thinly 
populated  districts  devoid  of  manufactories.  Such  districts, 
however,  are  not  often  to  be  met  with  in  the  vicinity  of  large 
towns,  and,  even  when  they  are,  we  must  expect  a  gradual  en- 
croachment upon  them,  from  the  natural  growth  of  the  neighbor- 
hood. 

One  excellent  criterion  of  quality  is  the  general  health  of  the 
communities  using  the  water.  Also,  the  order  of  fishes  which 
find  their  habitat  in  the  streams  of  a  given  district.  Little  dan- 
ger, for  instance,  would  be  apprehended  from  the  use  of  a  good 
trout  stream. 

Although  all  the  fresh  water  used  upon  the  earth  reaches  it 


72  PRACTICAL   HYDRAULIC    FORMULA. 

in  the  form  of  rain  which  is  first  drawn  up  by  evaporation,  chief- 
ly from  the  sea,  there  are  several  different  forms  under  which  it 
presents  itself  for  our  use.  The  first  broad  classification  of 
these  forms  would  be  that  which  divide.3  the  supply  into  surface 
water  and  ground  water.  By  surface  water  is  understood  the 
water  of  lakes,  ponds,  rivers  and  streams,  all  water  which  in  fact 
is  collected  directly  from  the  surface  of  the  earth;  and  by  ground 
water,  that  which  is  derived  from  wells  and  filtering  galleries,  and 
from  springs  when  taken  at  or  near  their  source. 

Each  of  these  classes  admits  of  much  subdivision,  but 
the  differences  will  be  principally  those  of  degree,  and  not  of 
kind. 

For  instance,  we  have  the  smaller  streams,  such  as  creeks, 
brooks,  etc.,  and  also  the  larger  ones,  rising  to  the  dignity  of 
rivers.  While  these  certainly  do  present  slightly  different  char- 
acters, still  their  main  difference  is  one  of  size.  Again,  though 
the  waters  of  lakes  and  ponds  differ  somewhat  from  each  other, 
and  from  those  of  streams  and  rivers,  still  they  are  only  the  col- 
lected products  of  these  latter,  which  they  consequently  greatly 
resemble. 

Ground  water,  proceeding  directly  from  the  earth,  offers 
more  distinctive  characteristics,  shared,  generally,  by  all  its  sub- 
classes. 

As  regards  the  relative  salubrity  of  water  drawn  from  minor 
streams  and  that  from  large  rivers,  it  would  seem  that  they 
stood  nearly  upon  a  par,  their  principal  difference,  as  already 
mentioned,  being  that  of  size.  At  first  sight  it  would  appear 
that  the  smaller  streams,  situated  near  the  headwaters  of  the 
larger  ones,  or  rivers,  would  possess  a  higher  degree  of  purity 
from  the  fact  of  the  water  being  collected  from  a  comparatively 
wilder  and  more  thinly  populated  district.  This  is  not,  however, 
of  necessity,  the  case,  because  the  river,  although  passing  through 
a  denser  population,  affords  by  its  greater  volume  a  greater  degree 


PRACTICAL   HYDRAULIC    FORMULAE.  73 

of  dilution.  The  true  criterion  in  respect  to  this  view  of  the 
question  would  seern  to  be  the  density  of  population  per  square 
mile  of  drainage  era,  together  with  the  proximity  of  its  center  of 
gravity  to  the  stream  itself.  The  further  the  hulk  of  the  popula- 
tion is  from  the  stream  or  river,  the  greater  the  chances  of  purifi- 
cation by  natural  filtration.  Large  rivers  are  apt  to  have  towns 
directly  on  their  banks,  which  drain  all  their  sewage  immediately 
into  the  stream. 

Large  rivers,  being  made  up  principally  of  the  smaller  streams, 
would  appear  to  form  the  general  average  of  all  their  feeders.  It 
must  be  borne  in  mind,  however,  that  besides  the  yield  of  the 
smaller  streams  the  river  is  partly  fed  by  direct  surface  wash 
from  its  immediate  banks,  thus  imparting  to  it  a  somewhat  modi- 
fied character,  distinct  from  that  of  the  smaller  brooks  emptying 
into  it,  and  causing  the  water  of  the  large  river  to  be,  as  a 
general  rule,  softer  and  warmer  than  that  of  its  small  feeders. 
Generally  speaking,  however,  in  the  large  river  you  get  all  the 
water  and  all  the  impurities,  thus  making,  as  already  stated,  a 
pretty  fair  general  average  of  the  whole,  while  of  the  smaller 
feeders  some  will  have  a  greater  and  some  a  less  degree  of  concen- 
tration of  impurities  than  the  average. 

There  is  another  point  worthy  of  note  as  regards  the  relative 
quality  of  the  water  taken  near  the  mouth  of  a  great  river,  or 
from  the  smaller  streams  near  its  source.  All  impurities  enter- 
ing such  streams  or  rivers  have  a  greater  chance  of  being  exter- 
minated by  oxidation,  by  the  lower  forms  of  organic  life  and  by 
fishes,  the  longer  they  remain  exposed  to  these  agencies.  Hence 
Hear  the  moutli  of  long  rivers  we  have  a  right  to  assume  that 
many  of  the  impurities  which  entered  near  the  headwaters  have 
been  destroyed  during  their  long  passage  toward  the  mouth.  On 
the  other  hand,  this  prolonged  sojourn  has  increased  the  proba- 
bility of  development  of  disease  germs  which  have  escaped  actual 
destruction.  The  question  then  comes  up  :  Had  we  better  take 
our  impurities  and  disease  germs  fresh  or  stale  ?  And  the  answer 


74  PRACTICAL    HYDRAULIC    FORMULA. 

would  probably  be  :  Fresh,  if  we  must  take  them  at  all,  and  can- 
not trust  to  time  for  their  destruction. 

There  is  another  point  of  difference  between  Che  two  classes 
of  streams,  which,  although  possessing  an  engineering  rather  than 
a  sanitary  character,  it  may  not  be  amiss  to  refer  to  here.  In  the 
case  of  the  small  streams  a  greater  necessity  will  generally  exist 
for  storage,  in  order  to  secure  a  uniform  supply,  while  gravity 
can  be  more  often  counted  upon  as  a  motive  power  than  in  the 
case  of  the  large  river,  where  storage  is  seldom  needed,  and  where 
pumping  is  almost  invariably  necessary  ;  a  notable  exception 
being  that  of  Washington,  D.  C.,  where  the  water  of  the  Potomac 
flows  into  the  city  by  gravity. 

Ground  water  may  be  drawn  from  shallow  or  deep  seated 
wells — the  latter  of  ten  improperly  called  artesian — from  galleries, 
or  directly  from  springs  at  the  point  where  they  burst  forth  from 
the  earth. 

Shallow  wells  are  supplied  by  the  rain  which  falls  and  soaks 
into  the  ground  in  their  immediate  vicinity.  In  seasons  of  much 
rain  the  level  of  saturation  is  comparatively  near  the  surface  ;  in 
seasons  of  drought  the  level  descends  as  the  water  gradually  drains 
off  to  the  nearest  valley. 

While  an  isolated  shallow  well  may  afford  water  of  excellent 
quality  and  considerable  relative  coolness,  such  wells  situated  in 
towns  and  villages,  or  even  when  located  near  the  cesspools  of 
solitary  dwellings,  constitute  what  upon  the  whole  must  be  con- 
sidered the  most  objectionable  supply  in  common  use.  Their 
hardness  and  saltness  when  compared  with  neighboring  springs  are 
a  good  indication  of  their  relative  contamination  by  human  ref- 
use. These  qualities  are  observed  to  increase  with  time,  and  the 
growth  of  the  village — a  striking  corroboration  of  what  has  been- 
advanced  above. 

Deep  seated  wells,  and  springs,  may  be  fed  by  rain  falling  on 


PRACTICAL    HYDRAULIC    FORMULAE.  7.-) 

far  distant  points.  The  water  from  these  wells  is  apt  to  be  im- 
pregnated with  earthy  salts,  and  therefore  to  be  hard,  frequently 
to  the  extont  of  unfit  ness  for  domestic  uses.  The  temperature  is 
apt  to  be  higher  than  that  of  shallow  wells. 

The  supply  from  deep  wells  is  more  abundant  and  steady  than 
from  shallow  ones,  the  volume  of  supply  being  more  dependent 
upon  depth  than  diameter  ;  indeed  increased  diameter  only  affords 
greater  storage  in  any  given  case. 

Ground  water  is  frequently  obtained  from  drains  or  filtering 
galleries,  or  lines  of  pipe  with  open  joints,  chiefly  located  near 
and  parallel  to  and  lower  than  rivers,  which  galleries  intercept 
the  water  flowing  through  the  ground  toward  the  river,  and 
which  probably  are  also  fed,  to  some  extent,  by  the  water  of  the 
river  itself,  leaching  back  to  the  drain,  gallery  or  pipe  line. 

Water  in  considerable  quantities  is  sometimes  collected  from 
springs,  and  conveyed  away  immediately  as  it  bubbles  up  from 
the  ground.  The  new  water  supply  of  Havana,  Cuba,  is  a  nota- 
ble instance  of  this,  where  some  four  hundred  springs,  furnishing 
over  five  millions  of  cubic  feet  per  24  hours,  have  been  collected 
about  ten  miles  from  the  city,  and  the  water  conveyed  in  an 
aqueduct  to  a  distributing  reservoir,  whence  it  is  delivered  to  the 
city  in  cast  iron  pipes.  Such  a  supply  seems  likely  to  be  the 
purest  that  can  be  obtained.  Nevertheless,  spring  water  is  apt 
to  be  somewhat  hard  from  the  amount  of  earthy^  salts  frequently 
held  in  solution. 

It  will  be  seen  from  the  above  that  the  question  of  the  rela- 
tive purity  of  different  classes  of  water  is  a  very  complicated  and 
uncertain  one,  not  admitting  of  a  general  solution,  but  involving 
the  consideration  of  a  great  number  of  special  cases.  Ordinarily 
the  choice  in  any  given  instance  is  very  limited,  most  towns  hav- 
ing but  few  sources  of  supply  to  select  from.  The  choice  is 
ordinarily  further  controlled  and  limited  by  questions  of  quan- 
tity and  cost,  so  that  it  seems  hardly  worth  while  to  consider  the 


76  PRACTICAL   HYDRAULIC   FORMULAE. 

subject  under  its  general  aspect  at  all,  but  simply  to  make  a  special 
study  of  each  special  case. 

QUANTITY  OF  WATER. 

Next  in  importance  to  quality  comes  the  question  of  quan- 
tity. It  will  be  observed  that  the  growing  tendency  is  to  increase 
the  amount  allotted  per  capita  per  diem.  It  is  found  io  be  neces- 
sary to  make  abundant  provision  for  the  future  growth  of  the 
town  to  be  supplied,  to  anticipate  an  increasing  individual  use  of 
water,  and  also  to  provide  for  the  yearly  increase  of  leakage,  con- 
sequent upon  the  gradual  deterioration  of  the  work  and  of  the 
house  plumbing.  This  latter  is  a  fruitful  though  often  overlooked 
cause  of  a  diminishing  supply. 

In  general  it  may  be  said  that  a  hundred  gallons  per  twenty- 
four  hours  per  capita,  to  be  consumed  in  ten  hours,  with  a  liberal 
allowance  for  future  growth  of  population,  is  a  safe  but  not  ex- 
travagant estimate.  We  frequently  hear  of  a  town  finding  that 
its  water  supply  has  become  inadequate ;  we  never  hear  of  one 
suffering  from  too  great  a  one.  The  control  and  diminution  of 
waste  are  now  occupying  a  great  deal  of  attention,  particularly  in 
England,  where  the  density  of  the  population  renders  strict  econ- 
omy necessary.  Frequently,  no  doubt,  the  best  and  cheapest  way 
of  increasing  a  deficient  water  supply  would  be  to  reduce  waste, 
by  the  use  of  meters  and  other  means  for  securing  the  co-opera- 
tion of  consumers. 

From  the  purely  engineering  point  of  view  the  principal  in- 
terest involved  in  the  hourly  supply  is  its  connection  with  the 
size  of  pipes  required  for  its  delivery.  A  hundred  gallons  per 
head  per  twenty-four  hours  if  delivered  in  ten  hours,  is  at  the 
rate  of  ten  gallons  per  head  per  hour,  or  about  0.00037  cubic 
foot  per  second. 

It  has  just  been  stated  that  quantity  is  secondary  to  quality, 
but  in  studying  a  water  supply  project  the  first  step  is  to  decide 
upon  the  quantity  necessary  or  desirable  to  obtain.  This  fact 


PRACTICAL    HYDRAULIC    FOliMULJE.  77 

being  settled,  the  question  will  naturally  follow,,  How  can  we  as- 
certain what  the  yield  of  a  given  stream  will  be  ?  One  way  is  by 
gaging,  and  this  should  be  always  done,  choosing  both  the  driest 
and  wettest  seasons  for  the  purpose.  But  it  is  evident  that  all 
this  takes  time,  and  even  a  year's  continuous  gaging  would  not 
be  considered  as  conclusive  in  any  case  where  the  demand  nearly 
approaches  the  probable  supply,  because  we  must  calculate  on  an 
occasional  year  or  two  of  very  exceptional  drought.  Another 
way  is  to  make  a  survey  of  the  area  which  drains  into  the  stream 
under  study,  above  the  point  at  which  it  is  proposed  to  take  the 
water.  This  area,  combined  with  the  rainfall,  known  or  assumed, 
aiid  a  general  knowledge  of  the  character  of  the  watershed,  fur- 
nish reliable  data  for  calculating  the  approximate  yield  of  the 
stream.  Here  again,  however,  we  are  confronted  with  the  neces- 
sity of  consuming  much  time,  for  although  the  survey  can  be 
rapidly  made,  the  records  of  rainfall  require  at  least  as  much  time 
as  does  gaging.  Fortunately  in  many  cases  we  can  make  pretty 
close  estimates  of  the  amount  of  water  probably  derivable  from  a 
given  area,  by  using  data  already  collected  for  neighboring  dis- 
tricts, and  at  any  rate  we  can  always  make  reasonable  assumptions 
when  once  we  know  the  number  of  square  miles  of  territory  which 
drain  into  our  stream — the  liability  of  such  assumptions  to  be 
correct  increasing  with  the  area,  for  a  large  area  is  less  subject  to 
special  variations  from  local  causes  than  a  small  one. 

The  average  yearly  rainfall  in  the  Croton  basin,  which  fur- 
nishes by  far  the  larger  part  of  the  water  supply  of  the  city  of 
Xew  York,  is  about  46  inches.  Long  experience  shows  that  in 
this  basin  each  square  mile  of  watershed,  or  drainage  area  may 
be  safely  counted  upon,  one  season  with  another,  to  furnish  one 
million  of  U.  S.  gallons  per  twenty-four  hours,  or  365  millions  per 
year.  On  the  other  hand,  a  precipitation  of  46  inches  gives  very 
nearly  800  millions  per  square  mile  per  year.  Hence,  in  the 
Croton  basin  about  40$  of  the  total  rainfall  is  found  to  be  avail- 
able for  water  supply. 


78  PRACTICAL    HYDRAULIC    FORMULA. 

It  must  be  borne  in  mind  that  the  above  yield  represents  the 
yearly  average,  which  may  easily  vary  forty  times  either  way  for 
any  given  shorter  period.  This  fact  establishes  two  important 
points  in  regard  to  water  supply.  First,  the  necessity  of  adequate 
storage  reservoirs  to  convert  this  yearly  average  into  a  daily  one  ; 
and  secondly,  the  necessity,  in  the  interest  of  safety,  to  give  these 
reservoirs  ample  overflows,  or  spillways,  in  order  to  provide  free 
escape  to  the  surplus  water  which  may  flow  into  them  in  immense 
volumes  during  freshets. 

These  considerations  bring  its  naturally  to  the  question  of  stor- 
age, a  most  important  and  by  no  means  simple  one.  The  amount 
of  storage  necessary  to  insure  a  regular  daily  supply  varies  of 
course  with  the  extent  of  the  watershed  in  proportion  to  the  de- 
mand. The  larger  the  area,  the  smaller  may  be  the  storage.  In 
some  exceptional  cases  the  supply  may  be  so  great  that  its  abso- 
lute minimum  yield  is  greater  than  the  maximum  demand,  and 
in  such  cases  no  storage  is  necessary.  On  the  other  hand  cases 
BO  unfavorable  may  possibly  occur  when  the  total  yearly  average 
is  needed,  and  this  leads  to  a  maximum  storage  capacity. 

Let  us  consider  such  a  case,  and  suppose  a  community  which 
requires  a  supply  of  10  million  gallons  a  day  from  a  drainage  area  of 
10  square  miles,  and  follow  the  course  of  events  through  an  entire 
year.  The  year  will  be  divided  into  three  periods:  The  period 
of  average  flow,  the  period  of  drought,  and  the  period  of  over- 
supply. 

The  period  of  average  flow  will  be  that  in  which  the  daily 
yield  of  the  stream  is  exactly  equal  to  the  daily  draft — in  our 
supposititious  case  10, 000,000  gallons.  The  period  of  drought  will 
be  that  in  which  the  yield  is  less  than  the  above,  and  the  period 
of  over-supply,  that  in  which  it  exceeds  it.  These  last  two  periods 
will,  of  course,  vary  in  intensity,  through  indefinite  gradations. 

In  order  that  the  storage  capacity  should  be  ideally  perfect, 
it  would  be  necessary  to  so  proportion  it  that  at  the  commence- 


PRACTICAL   HYDRAULIC    FORMULAE.  79 

incut  of  the  period  of  drought  the  reservoir  should  he  exactly 
full  and  of  capacity  sufficient  to  hridge  over  the  interval  between 
the  drought  and  the  commencement  of  the  period  of  average 
yield.  At  the  commencement  of  the  over-supply  or  freshet 
period,  the  reservoir  should  he  completely  empty,  and  of  capacity 
sufficient  to  receive  and  retain  all  the  surplus  water  until  the 
period  of  average  flow  was  again  reached.  Not  a  drop  should  or- 
dinarily escape  except  through  the  supply  pipes,  and  an  overflow 
or  spillway  should  he  unnecessary  except  to  provide  for  extraordi- 
nary contingencies,  such  as  cloudbursts,  etc. 

It  is  clear  that  such  an  ideal  state  of  things  is  impossible  of 
realization.  It  would  be  based  upon  a  regularity  of  regimen  that 
could  never  occur  except,  perhaps,  by  chance,  during  a  single 
year.  Even  in  periods  of  extreme  drought  (and  this  extreme  is 
a  variable)  there  would  be  some  water  flowing  in  the  stream, 
and  the  storage  reserve  would  need  to  be  drawn  upon  only  for 
the  difference  between  this  amount  and  the  daily  supply  ;  while 
during  freshets  the  amount  to  be  stored  would  be  reduced  by 
the  daily  supply  being  drawn  off.  Moreover,  besides  the  average 
intensity  of  droughts  and  freshets,  there  come  cycles  of  still 
greater  intensity,  all  of  which  circumstances  are  controlling 
factors  in  the  problem. 

lu  the  oase  assumed  the  only  way  to  secure  the  total  flow,  so 
that  none  shall  pass  to  waste  over  the  spillway  except  in  the  case 
of  a  cloudburst  or  of  some  other  phenomenon,  and  at  the  same 
time  to  provide  for  droughts  of  maximum  intensity,  is  to  con- 
struct a  reservoir  or  reservoirs  of  capacity  to  contain  the  total 
yearly  yield  of  the  stream,  and  to  commence  the  use  of  the 
supply  with  a  full  reservoir,  so  that  there  shall  always  be  a  year's 
supply  ahead. 

This  treatment  of  the  problem  is  certainly  a  heroic  one,  and 
has  probably  never  been  fully  carried  out  in  practice,  although 
the  city  of  New  York  is  reaching  well  on  toward  it,  in  the  vast 


80  PRACTICAL   HYDRAULIC    FORMULAE. 

storage  works  executed  and  contemplated  in  the  Croton  basin. 
Fortunately  so  unfavorable  a  case  as  the  one  assumed,  when  the 
total  yield  of  the  stream  is  needed,  rarely  presents  itself,  and  in 
the  majority  of  cases  there  is  an  excess,  more  or  less  considerable, 
of  the  supply  over  the  demand. 

The  solution  of  the  problem  of  storage  capacity  lies  between 
the  two  extremes  above  instanced,  in  one  of  which  no  storage  is 
necessary,  and,  in  the  other,  when  it  is  necessary  to  have  capacity 
for  the  yield  of  the  whole  year. 

It  is  evident  that  we  cannot  say,  d  priori,  of  any  proposed 
water  supply,  that  storage  for  so  many  days  will  be  necessary,  or 
sufficient,  without  knowing  at  least  approximately  the  total  yield  as 
well  as  the  desired  consumption.  In  cases  where  close  calculation 
is  needed,  as  when  it  appears  necessary  to  utilize  the  greater  part 
of  the  supposed  supply,  the  proper  course  to  pursue  is  to  ascer- 
tain, by  actual  survey,  the  drainage  area;  to  ascertain,  by  rational 
assumption  when  direct  observations  are  lacking,  the  average 
precipitation,  and  then  allow  from  one-quarter  to  one-third  of 
the  same  as  available,  backing  these  data  by  gagings,  as  com- 
plete as  may  be  possible,  of  the  stream,  and  calculate  the  storage 
capacity  accordingly. 

I  have  confined  myself  in  the  above  to  a  general  view  of  the 
principles  involved  in  planning  storage  reservoirs,  nor  do  I  think 
it  wise  to  enter  into  more  elaborate  calculations,  as  they  might  lead 
the  inexperienced  to  suppose  that  the  problem  really  admitted  of 
a  general  mathematical  solution.  Such  is  not  the  case,  and  un- 
less the  known  factors  point  clearly  to  self-evident  assumptions, 
great  caution  and  much  study  should  be  bestowed  upon  the  fixing 
of  the  data  on  which  the  design  oi  an  economical  and  satisfac- 
tory water  supply  is  to  be  based.  One  thing  is  certain  :  Except 
for  economical  reasons  there  is  no  danger  of  having  too  great 
storage  capacity.  I  do  not  happen  to  recall  an  instance  of  a  com- 
munity suffering  from  the  possession  of  too  much  stored  water, 


PRACTICAL   HYDRAULIC   FORMULAE.  81 

v.Jle  the  want  of  enough  of  it  is  proving  a  serious  trouble  to 
cities  and  towns  all  over  the  country. 

I  have  already  adverted  to  the  matter  of  adequate  spillways 
for  discharging  the  flood  waters  of  freshets.  There  is  no  uni- 
formity of  practice  for  the  dimensions  of  these  all-important  ad- 
juncts, and  it  is  probable  that  the  great  majority  of  those  now  in 
existence  have  been  proportioned  by  guesswork,  or,  "upon  gen- 
eral principles/7  This  is  all  wrong  ;  and  in  this,  as  in  all  other 
questions  of  design,  we  should  first  ascertain  what  conditions  our 
structure  will  have  to  fulfill,  and  then  dimension  it  accordingly. 

The  capacity  or  open  area  of  a  spillway,  is  made  up  of  its 
length  and  height  of  notch.  It  must  be  large  enough  to  pass  all 
the  water  of  extreme  floods  without  danger  of  over-topping  the 
dam.  Forty  times  the  average  flow,  or  40  million  of  gallons  per 
square  mile  and  per  24  hours — or  64  cubic  feet  per  square  mile 
per  second — is  none  too  liberal  an  allowance,  particularly  for 
earthen  dams.  For  this  amount  of  water  we  have  the  two  simple 
approximate  formulae  to  determine  the  length  and  depth  of  notch 
of  a  spillway,  the  depth  being  counted  from  the  level  of  the  lip 
of  the  dam  to  the  surface  of  still  water  in  the  reservoir  : 

L  =  20  V'Z  (0 

D  =  3  \~A±  C  (2) 

in  which  L  =  length  in  feet,  D  —  depth  in  feet,  A  =  area  of 
watershed  in  square  miles,  and  C  =  a  certain  additional  height 
above  the  water  in  the  reservoir,  depending  upon  the  character 
and  construction  of  the  dam.  If  we  should  wish  to  provide  for  a 
different  amonntof  water,  we  must  generalize  formula  (2),  writing: 

•y^ 

D  =  — -6—  x*VA  +  c.  O) 

in   which   Q  —  cubic   feet  per  second  per  square  mile. 

DAMS. 

Large  reservoirs  are  generally  formed  by  building  a  dam 
across  the  valley  of  the  stream  furnishing  the  supply.  Naturally, 


$3  PRACTICAL   HYDRAULIC   FORMULA. 

the  narrowest  point  is  chosen,  but  further  investigation  may 
prove  such  point  to  be  not  the  most  favorable  one.  A  solid 
foundation  is  the  first  requisite,  and  sometimes  firm  rock  is  found 
so  much  nearer  the  surface  at  a  point  where  the  valley  is  wider 
that  a  dam  built  there  would  be  actually  shorter  than  at  the  nar- 
rower point,  besides  saving  the  extra  excavation  to  get  down  to 
solid  bottom.  It  is  abundantly  worth  while  to  devote  considera- 
ble time  in  exploring  and  surveying,  before  fixing  definitely  upon 
the  location  of  the  proposed  dam. 

In  examining  the  character  of  the  foundations,  I  think  that  test 
pits  furnish  the  only  trustworthy  information.  At  great  depths, 
these  would  be  very  expensive,  and  recourse  is  generally  had  to 
drilling.  This  furnishes  good  indications  when  properly  inter- 
preted, but  also  has  occasioned  many  expensive  misconceptions  of 
the  ground.  The  test  pit  remains  the  only  sure  means  of  ascer- 
taining what  is  below  the  surface. 

The  character  of  the  ground  will  determine  the  class  of  dam 
which  should  be  built.  If  good  rock  bottom  is  to  be  found,  a  ma- 
sonry dam  will  be  the  best,  and  perhaps  not  much  more  expensive 
than  a  properly  constructed  earthen  one.  All  the  elements  of  a 
masonry  dam  are  more  fixed  and  precise  than  those  of  an  earthen 
dam  can  be,  so  there  is  less  necessity  for  piling  up  what  may  in 
reality  be  redundant  work,  to  provide  for  contingencies  which  we 
cannot  exactly  determine  qualitatively. 

Masonry  dams  may  be  divided  into  three  classes  ;  low,  medi- 
um and  high.  Although  the  lines  of  demarcation  are  somewhat 
vague,  we  may  class  all  dams  less  than  thirty  feet  high  as  low, 
those  between  thirty  and  sixty  as  medium,  and  all  those  above 
sixty  as  high.  Before  commencing  our  investigations  it  will  be 
extremely  useful  to  establish  certain  data. 

Calculation  shows  that  the  equation  of  equilibrium  of  a  dam 
with  vertical  faces  is  : 

Wx*  =  20.83  H*t  (4) 


PRACTICAL  HYDRAULIC    FORMULAE.  83' 

in  which  Jr=  weight  in  pounds  of  a  cubic  foot  of  the  masonry, 
H=  the  height  of  wall,  and  .T  =  its  thickness,  both  in  feet.  The 
weight  of  a  cubic  foot  of  water  is  assumed  at  62.50  pounds. 
From  this  equation  we  derive  : 

4.565  H 
x  = ——  .  (5) 

V  W 

These  equations  show  that  the  overturning  moment  varies  as 
the  square  of  the  height,  and  the  resisting  moment  as  the  square 
of  the  thickness,  and  the  square  root  of  the  density  of  the 
masonry;  while  the  value  of  x,  the  thickness,  varies  as  the  height, 
and  inversely  as  the  square  root  of  the  density,  of  the  wall  ;  that 
is  to  say,  from  (5)  we  deduce  : 

X    \~W 

=  4.565, 

H 

a  constant. 

If  we  assign  125  pounds  as  the  unit  weight,  or  weight  per 
cubic  .foot,  of  the  masonry,  we  find  : 

x  =  0.41  H.  (6) 

This  is  the  value  of  x  for  exact  static  equilibrium.  We  may 
obtain  whatever  factor  of  safety  we  wish  by  simply  multiplying 
the  square  of  0.41  by  such  factor  and  extracting  the  square  root 
of  the  product.  Thus,  suppose  we  wish  a  factor  of  2.5.  Oper- 
ating as  above,  we  find  : 

x  =  0.648  H.  (7) 

As  the  assumption  of  weight  is  somewhat  arbitrary,  we  may 
for  simplicity  write  (7)  thus  : 

2  H 

.=— :  ,8, 

that  is   to  say,  a  "  plumb  "  wall  to  resist  water  pressure  should 
be  twice  as  thick  as  one  to  resist  average  earth  pressure. 

However,  dams  are  not  built  plumb.  They  generally  have 
vertical  backs,  toward  the  water,  and  battering  faces.  The 
readiest  way  to  transform  a  vertical  or  plumb  wall  into  a  trape- 
zoidal one  of  equivalent  resisting  moment  is  to  follow  Vauban's 


84 


PRACTICAL   HYDRAULIC    FORMULAE. 


principle,  that  all  equivalent  walls  with  vertical  backs  have  the- 
same  thickness  at  one-ninth  of  their  height  from  the  bottom. 
This  rule  holds  very  closely  good  within  wide  limits. 

As  an  application,  let  us  take  the  case  of  a  wall  to  sustain 
water,  27  feet  high.  If  vertical,  its  thickness  should  be  18  feet 
for  a  factor  of  safety  of  2.50,  and  we  would  have  the  rectangle 
A  B  CD,  shown  in  Fig.  16.  Transforming,  according  to  Vau- 

A  E  IB 


K 


19.5 
20.25 -* 

FIG.  16. 

ban's  rule,  to  a  trapezoidal  section,  with  top  width  uf  ft  feel,  we- 
get  the  figure  A  E  F  C,  of  which  the  bottom  width  O  Pis  12>.50 
feet.  Verifying  the  comparative  stability  of  the  two  sections,  we 
find  that  of  the  trapezoidal  one  to  be  within  about  1J#  of  the 
rectangle,  while  its  area  is  nearly  30$  smaller. 

Such  a  section  as  A  E  F  (7  is  obviously  awkward,  presenting 
a  top-heavy  appearance,  from  the  redundant  thickness  of  its 
upper  half.  Some  such  section  as  A  E  H  G  C  would  be  pref- 


PRACTICAL    HYDRAULIC    FORMULAE. 


85 


•oruble ;  it   has  still  less  area  and  not  much  less  stability  than  the 
trapezoid  A  E  F  C. 

Passing  to  dams  of  medium  height,  let  us  take  for  an  example 
•one  54  feet  high  (Fig.  17).    The  proper  thickness,  if  vertical,  per 


40.50 


FIG.  17. 


•formula,  (8)  is  36  feet.  Transforming  the  rectangular  section 
into  a  triangular  one,  by  Vau ban's  rule,  we  obtain  the  right- 
angled  triangle  A  B  '  C  which  possesses  certain  interesting 
properties.  In  the  first  place,  the  triangular  section  has  only 


86  PRACTICAL    HYDRAULIC    FORMULAE. 

about  85$  of  the  stability  of  the  rectangle.  Secondly,  its  base 
will  always  be  three-quarters  of  its  height.  Thirdly,  the  result- 
ant of  its  weight  combined  with  the  thrust  of  the  water  (still 
assuming  the  specific  gravity  of  the  masonry  to  be  2)  will  always 
cut  the  base  about  11$  within  its  "middle  third";  while,  of 
course,  the  action  of  the  weight  alone  will  cut  the  base  exactly 
at  the  inner  extremity  of  the  middle  third. 

Evidently  such  a  section  is  impossible  in  practice,  because  it 
involves  a  top  width  of  zero.  Let  us  give  it  a  top  width  of  10 
feet,  with  a  face  batter  of  an  inch  to  the  foot  to  the  upper  part. 
This  batter  will  always  intersect  the  hypotenuse  of  the  triangle  afc 
a  distance  from  the  top  equal  to  one  and  a  half  times  the  top 
width,  whatever  that  may  be. 

This  composite  structure  has  a  resisting  moment  about  8fi 
less  than  that  of  the  rectangular  one  of  the  same  height,  and  base 
of  36  feet.  It  still  has  a  factor  of  safety  of  2.42,  with  the  above 
relative  densities  of  masonry  and  water,  while  the  section  is  about 
40#  less  than  the  rectangular  one.  Fig.  17  shows  all  dimensions, 
and  the  triangle  of  forces.  It  will  be  noted  how  the  addition  of 
the  upper  trapezoid  modifies  the  points  of  application  of  the 
pressures. 

In  regard  to  all  dams,  high  or  low,  we  may  lay  down  the 
leading  principle  that  the  line  of  pressure  should  always  pass 
within  the  middle  third  of  the  base,  especially  the  line  which 
corresponds  to  a  full  reservoir  ;  that  is,  the  line  which  is  the  re- 
sultant between  the  weight  of  the  dam  itself  and  the  thrust  of 
the  water.  In  very  high  dams  it  is  not  sufficient  that  this  condi- 
tion be  fulfilled  for  the  base  only  :  it  must  hold  good  also  for  any 
horizontal  bed  between  the  base  and  the  top,  because  in  such 
dams,  in  order  to  economize  material,  the  face  is  given  the  form 
of  a  convex  curve,  and  if  this  convexity  be  too  great  it  will  occur 
that,  while  the  base  may  have  a  satisfactory  width,  some  of  the 
upper  beds  parallel  to  it  will  not.  The  object  in  designing  a  high 
dam  is  to  give  the  section  such  a  form  that  it  shall  be  a  "section 


PRACTICAL    HYDRAULIC   FORMULA.  §7 

of  equal  resistance,'7  because  this  is  always  the  secrion  of  greatest 
economy. 

The  problem  is  further  complicated  in  the  case  of  very  high 
dams  by  the  fact  that  the  resistance  to  overturning  is  not 
the  only  thing  to  be  considered.  We  must  also  determine 
whether  the  area  of  the  lower  beds  is  sufficient  to  resist  the 
crushing  strain  brought  upon  them  by  the  weight  of  the  superin- 
cumbent mass.  In  making  this  investigation  it  is  obvious  that 
the  first  step  will  be  to  fix  a  proper  limiting  unit  strain,  or  ad- 
missible pressure,  per  square  foot  upon  the  masonry.  This  limit 
depends  upon  the  nature  of  the  material  used  and  also  upon  the 
views'  of  the  designer.  For  ordinarily  good  masonry,  15,000 
pounds  per  square  foot  would  be  considered  a  conservative  limit,, 
being  a  trifle  over  100  pounds  per  square  inch.  If  the  resultant 
of  the  pressures  cut  any  bed  exactly  in  the  middle,  we  could  ascer- 
tain the  pressure  per  square  foot  upon  such  bed  by  simply  divid- 
ing the  whole  weight  of  the  mass  resting  upon  it  by  its  length. 
But  when  the  resultant  moves  from  the  center,  the  strain  is  no 
longer  evenly  distributed  over  the  entire  bed,  but  is  intensified 
upon  that  portion  comprised  between  the  point  where  the  result- 
ant cuts  the  bed  and  the  nearer  extremity  of  the  same,  reaching 
its  maximum  intensity  at  the  extremity  itself,  or  as  we  should  say, 
at  the  nearer  toe.  The  investigation  of  this  varying  strain, 
which  increases  in  proportion  as  the  resultant  approaches  the 
nearer  toe,  is  somewhat  obscure,  and  rests  upon  assumptions  of 
somewhat  unsatisfactory  demonstration.  The  following  two 
formulas  may,  however,  be  accepted  as  reliable  approximations  to 
the  truth,  within  the  limits  occurring  in  ordinary  practice  : 

2  w 


4  W 
P  =  -  (L-1.5D),  (10) 

,£ 

in  which  : 

P=  pressure,  in  pounds,  per  square  foot. 


88  PRACTICAL   HYDRAULIC   FORMULAE. 

L  =  length  of  given  bed,  in  feet. 
W  —  weight,  in  pounds,  of  mass  above  given  bed. 

D  -  distance,  in  feet,  from  point  of  intersection  of  resultant  with  given  bed,  to 
nearer  extremity  of  same. 

L 

Formula  (9)   is  used   when   D  is  equal  to  or  less   than  — . 

L          3 
Formula  (10)  is  used  when  D  is  equal  to  or  greater  than — .  When 

L  3 

D  =  — ,  either  formula  gives  unit  strains  eoual  to  twice  the  total 

3 
weight  above  bed,  divided  by  its  length. 

We  have  then  the  three  following  conditions  which  the  pro- 
per section  of  a  high  masonry  dam  should  fulfill  :  First,  the  lines 
of  pressure  should  lie  within  the  middle  third  of  all  beds.  Sec- 
ondly, the  maximum  unit  strains  should  not  exceed  a  moderate 
fixed  limit.  Thirdly,  the  section  should  be  one  of  equal  or  nearly 
equal  resistance. 

Now  then,  in  the  light  of  what  has  been  already  established, 
let  us  feel  our  way  toward  the  proper  design  for  a  darn  160  feet 
high  fulfilling  the  above  three  conditions.  Let  us  assume  a 
density  of  masonry  double  that  of  water,  a  limiting  unit  strain  of 
15,000  pounds,  and,  as  is  usual  in  such  cases,  let  us  consider  a 
length  of  one  foot  of  dam,  so  that  the  area  of  our  section  in  square 
feet  will  represent  an  equal  volume  in  cubic  feet. 

Knowing  that  one  of  the  necessary  conditions  is  that  the  re- 
sultants shall  lie  within  the  middle  third  of  all  the  horizontal 
beds  which  we  may  suppose  to  divide  the  section,  we  feel  sure 
that  we  cannot  go  far  wrong  in  first  laying  down  the  right-angled 
triangle  ABC  (Fig.  18),  of  base  equal  to  three-quarters  of  the 
height,  or  120  feet  for  the  total  height  of  160  feet.  Desiring  a 
top  width  of  say  20  feet,  we  lay  off  the  same  from  A,  giving  the 
face  of  this  portion  of  the  section  a  batter  of  -j^-.  This  batter  we 
already  know  will  strike  the  hypotenuse  30  feet  vertical  from  the 
top.  Now  as  we  know  that  the  effect  of  placing  this  top  story 


PRACTICAL   HYDRAULIC   FORMULA.  89 

upon  our  triangle  will  be  to  draw  the  line  of  vertical  downward 
pressure,  due  to  weight  of  masonry  alone,  away  from  the  center 
of  gravity  of  the  triangle  A  B  G>  and  therefore  outside  of  the 
"  middle  thirds  "  on  the  water  side  ;  and,  also,  anticipating  a 


FIG.  18. 

little  the  knowledge  which  we  shall  presently  acquire,  we  give 
the  back  of  the  section,  from  a  point  80  feet  from  the  top,  an 
outward  flare  of  one  to  four,  which  is  shown  in  the  figure  hy  the 
small  triangle  D  E  B.  and  which  increases  the  total  width  of  base 
to  140  feet.  The  section  is  thus  divided  into  three  trapezoids, 


90  PRACTICAL    HYDRAULIC    FORMULA. 

respectively  30,  50  and,  80  feet  high,  with  corresponding  widths 
of  20,  223,  60  and  140  feet. 

Next  we  determine,  either  graphically  or  bv  simple  calcu- 
lation, based  upon  the  properties  of  similar  triangles,  the  points 
where  the  line  passing  through  the  center  of  gravity  of  the 
superincumbent  mass  cuts  the  imaginary  bed  D  F,  and  also 
where  it  cuts  the  same  when  shoved  forward  by  the  thrust  uf 
the  water,  acting  at  right  angles  to  it.  These  points  are  shown 
in  the  figure  to  be  respectively  at  19.35  ft.  and  25  ft.  from  D 
and  F.  The  figure  shows  also  the  intensity  of  the  strain  in  pounds 
at  these  points,  obtained  by  multiplying  the  total  area  above  D  F 
by  125,  the  assumed  weight  in  pounds  per  cubic  foot  of  masonry, 
although  the  calculations  were  actually  made  in  units  of  volume, 
for  the  sake  of  rapidity  and  ease,  counting  the  weight  of  a  cubic 
foot  of  masonry  as  1,  and  that  of  water  as  ^. 

We  next  proceed  in  the  same  way  in  regard  to  E  C.  Here 
we  have  first  the  point  where  the  line  passing  through  the  center 
of  gravity  of  the  entire  section  A  E  C  cuts  the  base  E  C,  55.80 
ft.  from  E,  and  which  corresponds  to  an  empty  reservoir  ;  and 
secondly,  the  point,  60.32  ft.  from  C,  where  the  resultant  of  the 
weight  of  the  section  A  E  G,  plus  the  weight  of  water  resting 
upon  the  inclined  surface  D  E,  combined  with  the  forward 
thrust  of  the  water  acting  under  a  head  of  160  ft.,  cuts  the  same 
base  E  C,  which  point  corresponds  to  a  full  reservoir.  The  in- 
tensities in  pounds  of  all  these  strains  are  shown  on  the  figure. 

Our  design  now  fulfills  one  of  the  imposed  conditions  .  The 
lines  of  pressure  lie  well  within  the  middle  third,  except  at  D, 
where  the  condition  is  not  so  binding.  It  remains  to  see  how 
it  complies  with  the  second  one.  For  this,  we  recur  to  our 
formulae  (9)  and  (10);  and  first  to  ascertain  the  unit  strain  at 
D.  For  this  we  employ  (9),  within  which  the  case  just  falls. 
Substituting  numerical  values,  we  have  : 

2  X  337500 

P  =  -  -  -  =  11628  Ibs.  per  sq.  ft. 

3  X  19.35 


PRACTICAL    HYDRAULIC    FORMULAE.  91 

For  the  strain  at  F  we  use  (10): 

4  X  337500 
P  =  -  —  (60  -  37.5)  =  8438  Ibs.  per  sq.  ft. 

3600 

Passing  to   E  C,  we   have  for  maximum  strain  at  E,  when 
reservoir  is  empty, 

4  X  1337500 
P  =  -  —  (UO  -  83.70)  =  15368  Ib8.  per  sq.  ft.  f  OF  THE 

19600  I  TJNIVERSI 

For  maximum  strain  at  C,  when  reservoir  is  full  :       x.o 

.  P  =  I*-™""*  <MO  -  90.48,  =  15033  .bs.  per  „.  ft. 

The  maximum  strains  are  given  on  the  plan  by  the  under- 
lined figures  at  D,  F,  E  and  C. 

Examining  our  design,  we  see  that  although  it  practically 
satisfies  the  first  two  conditions  demanded,  it  is  by  no  means  a 
section  of  equal  resistance,  for  the  strains  at  D  and  jPare  far  less 
than  those  at  E  and  C.  Evidently  the  upper  bed  D  F  is  too 
wide. 

As  a  further  step  in  our  tentative  process,  I  will  now  offer, 
Fig.  19,  a  section  suggested  by  Seiior  D.  E.  Boix,  in  his  excellent 
treatise  on  "  La  extabilidad  de  las  const  nice  tones  de  Main- 
posteria"  as  a  general  approximate  type  for  high  masonry  dams. 
Beginning  at  the  top,  the  skeleton  of  this  design  consists  of  a 
right-angled  triangle  A  B  Cot  base  equal  to  two-thirds  of  its 
height,  which  height  Seflor  Boix  makes  a  constant  of  24  meters, 
or  say  80  ft.  From  B  the  back  slopes  to  D  with  a  batter  of  $, 
and  from  C  to  K  with  one  of  f.  The  skeleton  of  the  design, 
therefore,  in  this  particular  instance  of  a  total  height  of  160  ft., 
consists  of  a  trapezoid  BCD  E  80  ft.  high,  140  ft.  wide  at 
bottom  and  53.33  at  top,  surmounted  by  a  right-angled  triangle 
80'ft.  high.  This  upper  triangular  part  is  a  constant,  for  all 
sections ;  the  variation  occurring  in  the  lower  trapezoidal  part, 
according  to  the  total  height  of  dam.  As  a  practical  detail,  the 
upper  part  is  surmounted  with  another  triangle,  giving  the  sec- 
tion a  proper  top  width.  I  have  assumed  a  top  width  of  20  ft., 
with  batter  J_,  to  correspond  with  previous  example.  These  di- 


92  PRACTICAL  HYDRAULIC   FORMULAE. 

mensions,  with  the  triangles  of  forces  and  strains  per  square  foot 
at  the  points  B,  C,  D  and  E,  are  shown  in  the  figure. 

This  may  be  considered  an  improvement  upon  the  previous  de- 
sign. Its  area  is  about  5$  less  and  there  is  a  much  better  distribu- 
tion of  strains.  Its  resisting  moment  is  a  little  less,  being  as  2.87  to 


155* 


FIG.  19. 

3.10,  but  the  coefficient  of  stability  is  sufficient.  Its  practical 
superiority  lies  in  the  5$  of  economy.  In  a  careful  final  study  of 
a  high  dam,  it  would  be  necessary  to  pass  a  greater  number  of 
horizontal  beds  through  the  section,  and  calculate  the  unit  strains 
at  each  extremity  of  each.  The  result  would  probably  lead  to  a 
more  pronounced  curve  oa  the  face,  with  a  corresponding  saving 
of  material.* 


*  It  is  well,  however,  that  the  upper  part  of  the  dam  should  have  a  greater  pro- 
portional strength  than  the  base,  because  it  is  exposed  to  a  greater  degree  of  wave 
action. 


PRACTICAL   HYDRAULIC    FORMULAE.  93 

An  example  of  a  well-proportioned  dam,  80  feet  high,  is  given, 
with  all  its  dimensions  and  the  triangles  of  forces,  in  Fig.  20. 
Calculations,  similar  to  those  used  already,  show  maximum  strains 
to  be  as  follows  :  At  E,  empty  reservoir,  8,677  pounds  per  square 

*  1-4* 


FIG.  20. 

foot;  at  F,  full  reservoir,  7,607.  At  G,  empty  reservoir,  9,250 
pounds  per  square  foot ;  at  //,  full  reservoir,  8,334.  The  water 
pressure  on  E  G  has  not  been  counted, 

In  designing  a  high  dam,   it  should  be  borne  in  mind  that 
too   much   confidence   is   not   to    be   placed  upon   the    formulae 


94  PRACTICAL   HYDRAULIC   FORMULAE. 

used  in  calculating  the  unequally  distributed  strains,  arid  that 
the  further  their  resultant  moves  from  the  center  of  the  beds  the 
less  they  are  to  be  depended  upon.  As  a  further  complication 
Sefior  Boix,  in  the  work  already  mentioned,  calls  attention  to  the 
fact  that,  instead  of  considering  only  the  vertical  component  of 
the  resultant  acting  upon  a  horizontal  bed,  we  should  consider  the 
resultant  itself  acting  upon  an  imaginary  bed  inclined  at 'right 
angles  to  it.  He  shows  that  this  gives  a  more  or  less  augmented 
unit  strain.  In  an  example  which  he  gives  of  a  darn  about  100 
feet  high,  the  pressures  thus  calculated  exceed  those  calculated 
upon  the  hypothesis  of  vertical  action  upon  a  horizontal  bed  by 
10$  to  17$.  It  would  greatly,  and  I  think  unnecessarily,  compli- 
cate the  problem  to  treat  it  in  this  manner,  for  the  weight  of  the 
superincumbent  mass  of  masonry  and  the  angle  of  the  resultant 
are  interdependent,  and  tedious  processes  by  trial  and  error  would 
be  needed  for  each  bed.  The  circumstance  is  merely  referred  to 
in  order  to  show  the  necessity  of  keeping  well. within  the  margin 
of  safety,  for  it  must  be  borne  in  mind  that  a  dam  .once  built 
cannot  readily  be  remodeled,  and  should  stand  intact  and  without 
material  repairs  as  long  as  the  town  does  which  depends  upon  it 
for  its  water  supply.  The  prototype  of  the  modern  high  masonry 
dam  is  that  across  the  valley  of  the  Furens,  near  Sc.  Etienne,  in 
France.  The  perfect  success  of  this  dam  is  no  proof  that  its 
section  is  suited  for  one  of  indefinite  length,  for  it  is  itself  very 
short,  and  wedged  in  between  the  rocky  sides  of  the  narrow  valley 
which  it  spans,  thereby  receiving  great  additional  strength  from 
these  lateral  supports. 

As  regards  the  plan  of  a  high  masonry  dam — that  is,  whether 
it  should  be  straight  or  curved,  with  the  convexity  up  stream — it 
cannot  be  said  that  a  curved  plan  is  necessary,  nor,  on  the  other 
hand,  can  it  be  denied  that  such  plan  is  an  element  of  strength, 
particularly  if  the  dam  be  short.  By  adopting  this  form,  in  case 
of  a  slight  movement  occurring  when  the  dam  comes  to  its  bear- 
ings under  pressure,  the  character  of  the  strain  will  be  always 


PRACTICAL   HYDRAULIC    FORMULAE.  95 

compressive  ;  while,  if  the  axis  be  straight,  any  forward  move- 
ment, however  slight,  will  produce  tensile  strains,  which  are 
always  to  be  avoided  in  unelastic  materials  like  masonry. 

Concerning  the  further  details  of  the  design  of  high  masonry 
dams,  they  should  stand  upon  a  base  or  pedestal  of  which  the  top 
is  level,  more  or  less,  with  the  surface  of  the  ground,  with  ample 
offsets  or  projections  beyond  the  toes  of  the  dam  proper,  particu- 
larly on  the  downstream  side.  The  excavation  should  go  through 
the  superincumbent  earth,  to  and  into  the  solid  rock.  The  sides 
of  the  base  should  be  vertical  and.  in  the  rock,  built  or  packed 
close  against  the  sides  of  the  excavation.  In  earth,  the  space  be- 
tween the  vertical  sides  of  the  base  and  those  of  the  excavation 
should  be  filled  up  solid  with  closely  compacted  and  puddled  ma- 
terial, so  as  to  leave  no  vacancies.  The  angle  which  the  outer 
slope  of  the  dam  makes  with  the  horizon  should  never  be  less 
than  45°,  so  as  to  avoid  sharp  and  weak  edges  at  the  points  of 
maximum  strain.  Whenever  sufficient  material  can  be  obtained 
from  the  excavation  or  from  borrow  pits,  a  sloping  bank  of  well- 
compacted  earth  should  be  placed  against  the  back  of  the  dam 
for  at  least  a  quarter  or  a  third  of  its  height  from  the  ground, 
and  protected  by  riprapping.  This  bank  has  a  double  object:  it 
not  only  impedes  leakage,  but  by  establishing  a  permanent  thrust, 
or  at  least  support,  against  the  back  of  the  dam  when  the  reser- 
voir is  empty,  it  diminishes  the  range  of  pressures  existing  be- 
tween a  full  and  an  empty  reservoir.  All  of  these  features  are 
shown  in  Fig.  21. 

Before  leaving  this  question  of  design,  there  is  a  recommen- 
dation which  I  think  it  well  to  make,  not  only  in  regard  to  high 
masonry  dams,  but  to  all  other  engineering  structures  as  well.  It 
is  this  :  After  carefully  proportioning  the  work  according  to  ap- 
proved methods  of  calculation,  take  a  good  look  at  the  drawing, 
and  see  if  it  LOOKS  right  ;  if  not,  there  is  probably  something 
wrong  in  the  calculations,  and  they  had  better  be  gone  over  to  see 
where  the  mistake  is. 


96 


PRACTICAL    HYDRAULIC    FORMULAE. 


Where  rock  foundation  is  not  obtainable,  the  best  kind  of 
dam  will  be  an  earthen  one,  with  masonry  core  or  center  wall. 
No  earthen  dam  can  be  considered  safe  that  is  not  provided  with 
such  wall,  carried  down  to  a  water-tight  or  comparatively  water- 
tight stratum.  The  masonry  center  wall  is  the  only  sure  and 
permanent  means  of  cutting  off  percolations  through  the  bank, 
and  for  this  reason  it  should  be  carried  well  down,  on  the  princi- 
ple that  the  worse  the  character  of  the  ground,  the  deeper  should 


Fro.  21. 

be  the  foundations.  It  should  also  be  deeply  imbedded  in  the 
sides  of  the  valley.  The  worst  kind  of  ground  is  a  loose  drift 
formation,  containing  many  large  cobblestones.  In  such  ma- 
terial the  wall  must  be  sent  "  'way  down,"  till  perhaps  there  is  as 
much  under  as  above  ground,  and  the  embankment,  particularly 
on  the  water  side,  carried  far  back  with  a  very  easy  slope,  some- 
times as  much  as  five  or  six  to  one.  Clay,  sand,  and  fine,  compact 


PRACTICAL  HYDRAULIC    FORMULAE.  97 

gravel  are  the  best  bottoms  to  build  on;  quicksand  is  also  excel- 
lent, when  lateral  escape  can  be  prevented,  as  it  usually  can  be 
except  when  very  near  the  surface. 

Not  only  does  the  masonry  center  wall  prevent  percolation,, 
but  it  also  affords  the  means  of  making  perfectly  secure  connec- 
tions with  all  the  accessories  of  the  reservoir,  such  as  culverts, 
piping,  gate  towers,  etc.  The  failure  of  many  earthen  dams  has 
been  due  to  water  following  along  outside  the  culverts  or  pipes  by 
which  water  was  drawn  from  the  reservoir,  a  circumstance  which 
cannot  occur  when  these  appliances  are  bonded  in  with  a  tighter 
even  partially  tight  masonry  wall,  extending  from  flank  to  flank 
of  the  valley,  and  carried  down  to  a  good  bottom.  t 

Although  this  wall  is  supported  on  both  sides  by  the  embank- 
ment, it  should  be  of  sufficient  thickness  to  withstand  any  un- 
balanced pressures  to  which  it  may  be  subjected.  Where  highest 
it  may  have  a  bottom  thickness  of  about  one-quarter  its  height, 
and  be  drawn  in  at  the  rate  of  about  an  inch  to  the  foot,  but 
preferably  by  offsets  rather  than  batter.  This  would  always  give 
a  top  width  equal  to  one-twelfth  the  height,  which  might  some- 
times lead  to  unsuitable  dimensions,  requiring  modification.  The 
top  should  be  carried  up  at  least  as  high  as  the  level  of  highest 
water  in  the  reservoir. 

The  spillway  should  be  proportioned  according  to  the  prin- 
ciples already  laid  down,  and  if  possible  some  natural  depression 
should  be  found  back  of  the  dam,  by  which  the  overflow  may  be 
passed  over  to  another  valley,  or  into  the  same  one,  lower  down. 
Failing  this  condition,  a  massive  masonry  spillway  and  apron 
must  be  built,  generally  in  the  axis  of  the  stream,  either  .with 
curved  face,  like  the  Croton  or  Bronx  River  dams,  or  stepped, 
like  those  of  the  Scran  ton  Gas  &  Water  Co.,  at  Scranton,  Pa. 
I  am  inclined  to  prefer  this  latter  mode  on  its  engineering  merits, 
and  apart  from  the  fact  of  its  greater  cheapness.  The  section  of 
the  spillway  will  be  proportioned  according  to  the  principle.? 
already  laid  down  for  high  dams,  but  it  should  be  even  more 


98  PRACTICAL   HYDRAULIC    FORMULAE. 

massive,    for   it   has  to  stand  the  impact  of  fulling1  water.     Its 
width  at  its  foot  should  be  equal  to  its  height. 

In  regard  to  the  embankment,  it  is  ordinarily  specified  that 
this  should  be  made  of  selected  material.  This  is  certainly  rec- 
ornmendable,  and  the  best  material  the  site  affords  should  be 
used  in  preference,  but  it  is  almost  impossible  to  compel  con- 
tractors to  sort  the  material,  and  moreover  really  good  stuff  is  not 
always  within  reach,  in  which  case  the  volume  put  in  tire  embank- 
ment must  be  increased,  to  compensate  in  a  measure  for  its  lack 
of  quality.  The  surface  of  the  bank,  on  the  water  side,  should  be 
well  riprapped,  and  the  lower  slope  sodded  or  seeded  to  grass. 
The  bank  should  be  kept  wet  by  sprinkling  while  being  put  in, 
and  should  be  brought  up  in  horizontal  lifts,  and  not  made  from 
a  dump,  like  a  railroad  embankment.  It  should  be  compacted  as 
it  goes  in  ;  ordinarily  the  travel  of  the  carts,  wagons  and  scrapers 
used  will  be  sufficient  for  this  purpose  in  connection  with 
thorough  sprinkling.  The  area  covered  by  the  embankment, 
particularly  on  the  water  side,  should  be  carefully  "floated  off," 
grubbed  and  plowed,  so  that  the  material  of  the  embankment 
shall  come  in  contact  with  clean  earth.  It  is  also  well  to  dig 
cross  ditches,  parallel  to  the  center  wall,  to  secure  a  still  further 
"bonding  of  the  surface  of  the  ground  with  the  embankment. 
These  ditches  must  be  carefully  filled  and  tamped  with  the 
material  used  in  the  embankment.  An  excellent  form  for  the 
inside  or  water  side  of  an  embankment  is  a  series  of  slopes  and 
berms,  forming  one  or  more, terraces.  Working  from  berm  to 
berm  gives  a  good  opportunity  to  carry  the  work  up  level. 

The  best  way  to  draw  the  water  from  the  reservoir  is  by 
means  of  cast  iron  piping  running  through  the  center  wall  and 
terminating  on  the  water  side  in  a  tower,  built  in  with  the  center 
wall,  and  containing  grooves  for  the  reception  of  stop  plank. 
The  pipes  should  lead  on  the  land  side  into  an  easily  accessible 
gatehouse,  also  built  in  with  the  center  wall,  and  each  line 
should  be  provided  with  two  good  gates  or  valves,  the  inner  one 


PRACTICAL   HYDRAULIC   FORMULA.  99 

to  be  kept  always  open  or  partly  open,  and  the  outer  one  used  for 
current  operations.  Then,  should  the  latter  get  out  of  order, 
the  inner  gate  can  be  closed,  and  the  necessary  examination  and 
repairs  effected.  It  is  a  good  plan  to  build  iron  eye-beams  into 
the  walls  of  the  gatehouse  directly  over  the  gates,  so  that  in.  case 
of  wishing  to  remove  them  the  necessary  overhead  purchase  can 
readily  be  applied.  These  ideas  are  not  hypothetical,  but  have 
been  satisfactorily  carried  out  in  the  system  of  storage  reservoirs 
built  for  the  Scranton  Gas  &  Water  Company,  already  referred  to. 

It  will  now  be  well  to  make  some  remarks  respecting  the 
various  classes  of  work  embraced  in  the  construction  of  dams  and 
reservoirs.  In  such  structures  a  good  deal  of  concrete  is  generally 
employed,  especially  in  foundations  and  subfoundations,  for 
which  purpose  it  is  the  best  material  that  can  be  used.  Great 
care,  however,  must  be  taken  in  its  preparation  and  placing,  for 
there  is  perhaps  a  greater  difference  between  good  and  bad  in  con- 
crete than  in  any  other  kind  of  masonry.  Various  proportions 
have  been  recommended,  and  we  may  lay  down  as  a  general  prin- 
ciple, that  the  smaller  the  stones  and  the  larger  the  percentage  of 
cement  used,  the  more  water-tight  will  be  the  resulting  concrete. 
I  have  obtained  excellent  results  using  fine  ground  Portland 
cement  and  thorough  mixing,  with  one  of  cement,  three  of  clean 
sand,  and  six  of  broken  stone.  In  important  work  I  should  no'1 
care  to  use  a  less  rich  mixture  than  this,  which,  as  I  may  add  in 
passing,  was  that  employed  by  the  late  General  Gilmore  in  the  re- 
building of  Forts  Sumter  and  Moultrie,  in  Charleston  Harbor, 
with  the  exception  of  substituting  Rosendalefor  Portland  cement. 
A  sufficient  quantity  of  water  should  be  added  to  bring  the  mass 
to  a  very  decided  degree  of  moisture,  while  not  drowning  it  with 
an  excess. 

The  sand  and  cement  should  be  thoroughly  mixed  dry  ; 
then  the  stones,  previously  wetted,  are  added  and  the  whole  mass 
thoroughly  mixed,  water  being  added  from  time  to  time  till  it  is 
brought  to  the  proper  consistency.  Always  keep  the  bed  thin 


100  PRACTICAL   HYDRAULIC   FORMULA. 

and  avoid  getting  the  material  into  heaps,  which  prevents  proper 
mixing.  If  a  mixing  machine  is  used,  the  sand  and  cement 
should  still  be  mixed  dry  hy  hand  before  putting  into  the  machine. 
If  the  whole  process  is  done  by  hand,  there  should  be  turned  out 
about  two  cubic  yards  of  concrete  in  the  work  per  day  of  ten 
hours,  per  man  all  told,  including  all  employed  in  mixing,  placing 
and  tamping. 

This  is  about  as  much  yardage  as  an  ordinary  gang  of  stone 
masons  and  helpers  will  do,  laying  up  first  class  hydraulic  rubble, 
and  one  of  the  principal  reasons  for  concrete  being  cheaper  than 
rubble  is  the  fact  of  its  being  done  by  a  cheaper  class  of  labor- 
Indeed  contractors  generally  bid  entirely  too  low  on  hand  made 
concrete,  as  compared  with  rubble,  with  the  result  that  they  en- 
deavor afterward  to  do  as  little  of  it  as  possible,  trying  to  get 
rubble  substituted  for  it.  Tamping  should  be  carried  on  until 
the  mass  assumes  a  jelly  like  consistency,  or  at  least  till  moisture 
is  brought  to  the  surface.  This  is  a  sine  qua  non,  which  inspec- 
tors must  insist  on.  Prolonged  tamping  will  frequently  bring  up 
the  water  from  a  piece  of  concrete  which  appeared  to  be  hope- 
lessly dry  when  put  in.  Bear  in  mind,  that  although  too  much 
water  is  bad,  too  little  is  very  much  worse.  As  a  general  rule  the 
greater  the  percentage  of  broken  stone,  the  more  thorough  must 
be  the  mixing  and  tamping.  If  the  stone  be  hard  and  sharp  the 
soncrete  will  gain  by  giving  it  all  that  it  will  carry.  Concrete 
may  very  easily  be  made  too  rich  in  mortar  by  carrying  out  the 
erroneous  idea  that  the  smaller  the  percentage  of  stone  the  better. 
After  concrete  has  been  placed,  let  it  remain  undisturbed  and  be 
kept  moist  by  const^^t  sprinkling  for  as  long  a  time  as  possible. 

The  stone  masonry  used  in  the  construction  of  hydraulic 
work  is  special  in  its  character,  and  requires  therefor  special  care 
in  its  execution.  This  fact  should  be  fully  explained  to  prospec- 
tive bidders  before  they  send  in  their  proposal?,  or  there  will  surely 
be  remonstrances  when  the  work  begins.  One  principal  source 
of  contention  is  in  regard  to  the  size  of  the  stones  used.  The 


PRACTICAL   HYDRAULIC   FOhMULJE.  10 1 

work  goes  on  so  very  much  quicker  by  the  use  of  the  largest 
stones  the  derrick  can  lift,  that  both  engineers  and  contractors 
are  tempted  to  get  in  as  many  as  possible.  I  think,  however, 
that  the  fact  remains  the  same  that  a  more  water-tight  wall  can 
be  built  with  small  stones.  "  Small  stones"  must  be  taken  in 
a  relative,  and  not  an  absolute,  sense  in  this  connection  ;  near  the 
foot  of  a  thick  wall  larger  ones  can  safely  be  used  than  in  the 
thinner  upper  portions;  and  in  hauling  and  distributing  stone  on 
the  ground,,  contractors  should  be  warned  to  keep  the  smaller 
ones  handy  for  the  top. 

Great  care  must  be  taken  in  bedding  and  jointing  the  stones. 
They  should  be  laid  on  the  natural  bed,  the  best  and  flattest  bed 
down,  and  should  be  made  to  swim  on  their  beds ;  i.  e.,  they 
should  be  susceptible  of  being  swayed  from  side  to  side  with  a 
bar  after  they  are  laid  and  before  being  spalled  up.  Xo  stone 
should  rock  when  stepped  on.  When  work  first  begins,  stones, 
after  being  bedded  and  pronounced  all  right  by  the  foreman, 
should  frequently  be  raised  again,  so  that  he  may  see  that, 
after  all,  there  are  many  blank  patches  on  the  bed  of  the  stone 
where  it  did  not  come  in  contact  with  the  mortar.  All  stones, 
particularly  small  ones,  should  be  hammered  down  on  their  beds. 
One  advantage  of  using  large  stones  is  that  they  bed  themselves  to 
a  great  extent  by  their  own  weight.  All  the  joints  should  be 
carefully  made  up,  the  invariable  rule  being,  throughout  the 
wall,  that  there  shall  be  no  vacant  spaces,  but  that  all  that  is  not 
stone  must  be  mortar.  To  this  end,  stone  should  not  be  laid  too 
close  together,  but  ample  space  afforded  for  getting  the  trowel 
all  around  them  so  as  to  cut  the  mortar  well  under  and  into  the 
beds  and  joints.  When  the  joints  are  narrow,  they  should  be 
filled  with  mortar,  and  spalls  driven  in.  as  many  as  the  joint  will 
hold,  without  crowding,  the  only  limit  being  that  in  no  case  shall 
there  be  stone  to  stone.  Also  be  very  careful  that  after  a  stone 
has  been  well  bedded  it  shall  not  be  lifted  from  its  bed  by  wedg- 
ing in  toa  many  big  spalls  under  it.  One  of  the  tests  of  a  good 


102  PRACTICAL,   HYDRAULIC   FORMULAE. 

mason  is  the  way  he  makes  up  his  joints,  fitting  in  all  the  spalls 
he  can.  Ordinarily,  the  mason  finds  it  less  trouble  to  fill  up 
the  joints  with  loosely  thrown  in  mortar.  This  is  against 
the  interests  of  both  contractor  and  company.  Needless 
to  say  that  the  work  must  be  well  bonded,  breaking  beds  as  well 
as  joints.  Particular  attention  must  be  paid  to  the  mortar,  see- 
ing that  the  sand  and  cement  be  thoroughly  mixed  dry,  till  it  pre- 
sents a  uniform  color,  without  streaks  of  sand  and  cement.  Good 
average  proportions  are  1  cement  and  2  sand  for  Rosendale,  and 
1  cement  and  3  sand  for  Portland.  The  greater  the  proportion 
of  sand  the  more  thorough  must  be  the  mixing.  Guard  against 
having  the  mortar  too  wet,  but  let  it  always  be  worked  up  with 
the  trowel  to  a  soft  pudding  in  the  work.  Masons  frequently  call 
out  that  the  mortar  is  too  dry,  when  what  it  wants  is  simply  more 
tempering  to  bring  out  the  moisture.  The  mason  should  be  con- 
stantly tempering  up  his  mortar,  and  when  he  has  large  quanti- 
ties on  hand  he  must  call  on  his  helper  to  do  the  same  with  his 
shovel.  The  more  you  work  up  and  turn  over  fresh  mortar — 
particularly  when  Portland  is  used— the  better  the  results.  As  to 
quantity  of  wall  laid  up,  per  derrick,  I  think  that  each  double 
drum  steam  derrick  of  proper  sweep,  well  tended  and  constantly 
fed  with  materials,  should  be  good  for  from  30  to  60  or  even 
more  cubic  yards  per  day  of  ten  hours.  These  limits  are  rather 
elastic,  but  perhaps  it  would  not  be  fair  to  draw  them  closer,  so 
much  depending  upon  the  size  and  character  of  the  stones  used. 
I  have  frequently  timed  a  derrick,  to  see  the  number  of  trips 
made  in  a  given  time,  but  should  be  loth  to  fix  a  standard.  If 
we  allow  five  minutes  to  a  trip,  we  should  have  one  hundred  and 
twenty  per  day.  These  trips  convey  not  only  the  larger  stones  to- 
be  set  by  the  derrick  in  the  work,  but  also  all  the  mortar  and 
spalls  needed  for  bedding  and  jointing.  The  governing  factor,  I 
think,  will  prove  to  be  the  size  of  the  stones  set  by  the  derrick, 
for  it  will  set  a  large  one  about  as  quick  as  a  small  one.  One 
thing  the  contractor  should  closely  watch  in  his  own  interest, 


PRACTICAL   HYDRAULIC   FORMULA.  103 

and  that  is,  that  a  derrick  should  never  stand  idle.  If  it  is  not 
constantly  in  motion  he  should  at  once  find  out  what  is  the  mat- 
ter. Probably  it  is  under-manned  at  one  end  or  the  other,  and 
the  gangs  need  to  be  increased.  Strange  to  say,  in  regard  to  this 
question  of  rapidity,  that  the  better  the  work  is  done  the  quicker 
it  is  done,  because  done  systematically,  and  every  stroke  tells. 
No  work  lags  so  much  as  slovenly  work. 

One  word  more  regarding  large  stones  :  When  used,  they 
must  always  be  swung  in  place  with  the  derrick,  and  never  barred 
nor  rolled  to  their  bed  over  fresh  work.  This  must  be  insisted 
on,  or  the  bond  of  the  mortar  will  be  surely  destroyed  by  jarring 
and  displacing  the  stones  last  laid. 

In  dry  riprapping,  as  in  concrete,  the  best  results  attend  the 
use  of  stones  of  various  dimensions,  so  disposed  that  the  per- 
centage of  voids  shall  be  as  small  as  possible. 

When  laying  masonry  in  wet  excavations  see  that  the  pump- 
ing sump  is  sunk  below  the  bottom  course  of  masonry,  so  that 
the  pumps  shall  not  draw  the  mortar  out  of  the  work,  and  placed 
outside  of  the  area  to  be  built  over.  This  is  a  very  important, 
though  almost  always  neglected,  point.  In  backfilling  an  exca- 
vation after  the  masonry  has  been  built  in  it,  see  that  the  earth 
is  well  compacted,  dead  against  the  wall,  so  as  to  leave  no  va- 
cancy between  it  and  the  sides  of  the  excavation.  This  is  also 
an  important  and  generally  neglected  matter. 

•When  work  of  this  kind  is  undertaken  it  is  indispensable 
that  the  principal  engineer  should  give  his  close  personal  atten- 
tion to  all  the  details,  particularly  at  the  commencement,  because 
both  the  contractors  and  the  inspectors  must  be  not  only  told, 
but  shown,  what  is  required  to.  be  done.  Later  on,  when  it  may 
be  impossible  for  him  to  give  his  undivided  attention  to  the  exe- 
cution  of  the  work,  he  should  make  frequent  and  prolonged  visits 
to  it,  at  irregular  times.  It  is  no  reflection  on  the  honesty  or 
ability  of  the  contractors  and  engineering  staff  to  say  that  this  is 


104  PRACTICAL   HYDRAULIC  FORMULAE. 

indispensable  to  secure  good  results  ;  it  is  merely  another  way  of 
saying  that  if  the  chief  neglects  his  duty  he  cannot  expect  that 
the  others  will  properly  perform  theirs. 

One  of  the  principal  engineering  difficulties  which  dam-build- 
ing presents  is  what  to  do  with  the  water  while  the  work  is  in 
progress,  and  particularly  while  the  foundations  are  being  put  in. 
When  dealing  with  a  powerful  stream  the  problem  assumes  formi- 
dable proportions.  Since  it  is  indispensable  that  all  dams  should 
have  a  capacious  blow-off  culvert,  at  or  near  the  bottom  of  the 
reservoir,  I  think  the  best  way  will  often  be  to  get  this  culvert 
and  all  its  appurtenances  in  first,  so  that  the  stream  may  be  di- 
verted into  it,  and  thus  give  no  further  trouble  in  average  stages 
of  the  river.  Should  the  season  of  freshets  intervene  when  the 
culvert  may  not  be  able  to  handle  all  the  water  coming  down  to 
it,  the  work  must  be  so  prepared  that  the  flood  may  sweep  over  it 
without  damage.  Much  forethought  is  necessary  in  such  cases, 
and  more  or  less  risk  can  hardly  be  avoided  ;  the  important  point 
to  reduce  it  to  a  minimum. 


Another  difficulty  occurs  when  springs  are  encountered  in 
the  foundation  pits.  These  may  be  sometimes  dealt  with  by  pro- 
viding a  temporary  escape  for  them  through  the  masonry,  to  be 
closed  afterward  when  the  surrounding  masonry  has  thoroughly 
set.  They  can  also  sometimes  be  overmastered  by  pumping,  but 
the  ingenuity  of  all  hands  is  frequently  taxed  to  devise  means 
of  get-ting  rid  of  them,  and  the  firmness  of  the  engineer  will 
also  often  be  severely  tested  in  refusing  to  allow  the  work  to  be 
commenced  without  adequate  preparation,  or  to  permit  a  defec- 
tive foundation,  through  which  a  current  of  water  is  passing,  to 
remain  and  receive  its  superstructure  while  in  that  condition. 

The  above  comprise  some  of  the  most  essential  points  to  be 
observed  in  dam-building,  but  it  is  impossible  to  cover  the  whole 
ground,  and  the  engineer  intrusted  with  such  work  will  find  all 
his  experience  and  resources  called  into  constant  requisition. 


PRACTICAL   HYDRAULIC   FORMULAE.  105 

One  governing  principle  should  never  be  lost  sight  of  in  de- 
signing such  structures.  While  thoroughly  good  work  should 
be  exacted  in  the  smallest  detail,  yet  the  design  of  the  dam  should 
be  such  as  to  provide  for  disaster  arising  from  overlooked  defects 
of  workmanship  or  from  unforeseen  contingencies,  in  such  a  way 
as  to  prevent  or  at  least  to  minimize  the  consequent  destruction. 
In  the  case,  for  instance,  of  a  cloudburst  gorging  the  spillway  of 
.an  earthen  dam,  when  the  absence  of  the  guardian  has  prevented 
the  relieving  blow-off  culvert  from  being  opened  in  time,  there 
is  almost  always  some  point  at  which  the  breach  can  be  made 
with  the  least  bad  results,  and  the  design  should  favor  its  occur- 
rence at  this  point  rather  than  in  less  advantageous  ones.  I  may 
add  that  the  existence  of  a  stout  masonry  center  wall  is  always  a 
tower  of  strength,  and  may  frequently  prevent  a  catastrophe 
should  the  dam  be  overtopped.  "  And  even  if  the  worst  should 
'"  occur,  and  the  center  wall  be  breached,  time,  that  priceless 
"element  in  such  cases,  would  be  gained,  and  the  catastrophe 
"  greatly  modified.  For  the  only  difference  between  the  harm- 
"  less  emptying  of  a  reservoir,  and  a  Johnstown  disaster,  is  one 
"of  time.""* 


*  Discussion  of  Mr.  Desmond  Fitz  Gerald's  paper  on  "  Rainfall,  Flow  of  Streams, 
and  Storage,"  Transactions  of  the  American  Society  C.  E.,  Vol.  XXVII.,  page  295, 
which  see  also  for  further  remarks  about  masonry  center  walls  and  capacities  of 
spillways. 


NOTES  TO  PARTS  I.  AND  II. 

Page  18.  —  It  will  be  observed  that  (1)  may  be  written  : 

v     .  /  J>"x"g 

=  V  CXL 


This  shows  that  V  increases  with  the  square  root  of  the 
diameter  and  of  the  head,  and  diminishes  with  the  square  root  of 
the  coefficient  and  length. 

If  diameter  d  be  given  in  inches,  then  to  obtain  Fin  feet,, 
multiply  C  by  12.  Thus  : 


V  =  ,v 

L 

Page  SO. — The  general  form  of  an  adfected  quadratic  is: 

*2  <±  a  x  =  b 


Whence  : 


Page  31.  —  Throughout  this  book  proportions  are  written  in 
the  fractional  form,  as  being  the  most  convenient.     The  propor- 

tion -  -  =  -,  is  read  :  "h  is  to  22  as  50  is  to  54.87. 

<i<>        54.  o7 

Page  42.  —  All  the  relations  between   the  different  elements 
of  two  long  pipes  may  be  expressed  in  a  single  equation  : 

D'*.H'  .Q*.L.  0 


~ 


Assuming  a  mean  common  value  for  (7  and  C': 

I?5  .H'  .Q*.L 


WATER  SUPPLY  ENGINEERING,  107 

The  most  useful  outcome  of  the  above  is  when  two  pipes 
have  the  same  length  and  head  ;  that  is,  when  L  =  L'  and  H  = 
//'.  Then  : 


Q 

That  is,  other  things  being  equal,  the  quantities  discharged 
by  two  pipes  are  as  the  square  roots  of  the  fifth  powers  of  their 
diameters. 

It  is  generally  easier  and  more  satisfactory  to  ascertain  what 
we  want  to  know  about  a  given  pipe  by  direct  calculation,  rather 
than  to  deduce  it  from  the  known  elements  of  another  pipe,  by 
means  of  the  relation  (a). 

Page  47.  —  The  great  practical  importance  of  the  formula} 
on  this  page  has  not  been  brought  into  sufficient  prominence  in 
the  original  text.  A  more  thorough  elucidation  will  be  given 
now. 

Equation  (3),  page  18,  can  be  written  thus  : 


Q=      /0.616J'.g 

r      c.x, 

Consulting  the  table  of  coefficients  on  the  same  page,  we  see 
that  those  for  rough  pipes  from  8  to  48  in.  in  diameter  vary 
from  0.00068  to  0.00062.  These  coefficients  do  not  greatly 
differ,  and,  moreover,  their  square  roots  only  enter  the  formulas. 
If,  then,  for  pipes  of  the  above  diameters  we  take  a  uniform 
value,  G  =  0.000616,  the  above  formula  reduces  to  : 

Q  =  i/l——JL  <6) 

L 

H 

Furthermore,  since — is  a  ratio,  we  may   always  reduce  If 

to  its  value  when  L  =  1000  ;  that  is  to  say,  we  can  establish  the 
relation  : 

h        H 

1000  ~  L 


108  WATER  SUPPLY   ENGINEERING. 

in   which   h  equals   the   fall,  or   head,  per    1000.      Therefore, 

H  h 

replacing —  in  (b)  by  its  equal ,  we  have  : 

//  1000 

Q  —  \' 


Always   recollect  that  li  represents  the  head  per  1000,    as 
against  //,  which  represents  the  total  head  in  the  total  length. 
The  relation  (c)  may  be  generalized  thus  : 


Again,  from  table,  page  18,  we  see  that  the  value  of  C,  for 
rough  pipes  from.  3  to  6  in.  in  diameter,  varies  from  0.00080  to 
0.00073.  Also,  equation  (3),  on  same  page,  may  be  written  : 


0.785  X 0.785  />•  .  H 

g  =  |/  - 

C  .L 

Adopting,  therefore,  a  mean  value  of  0.000785  for  C,  for  pipes 

H         h 

of  above  diameters,  and  reducing  —  to ,  as  before,  we  have  : 

L        1000 


Q  =    V  0.785  D4  h  (e) 

and  generalizing, 

Q3 

=  0.785  (/) 

hD* 

For  smooth  pipes,  from  8  to  48  in.  diameter,  (d)  becomes  : 
#2 

—  =  2  (0) 

hD* 

For  smooth  pipes  from  3  to  6  in.  diameter,  (/)  becomes  : 

Q* 

=  1.57  (h) 

hD* 

From  the  above,  we  see  that  the  discharge  of  a  smooth  pipe 
is  1.40  times  that  of  a  rough  one  of  same  diameter,  while  that  of 
a  rough  one  is  0.70  times  that  of  a  smooth  one. 


WATER  SUPPLY   ENGINEERING,  109 

Observing  that  velocity  is  always  equal  to  quantity  dis- 
charged divided  by  area  of  pipe,  we  have  for  the  velocity  of  flow, 
in  feet  per  second,  for  rough  pipes,  from  8  to  48  in.  diameter  : 

v=  1.27  v~Dh  W 

For  those  from  3  to  6  in.  diameter  : 


For  smooth  pipes  respectively  : 

V  =  1.80    VD~h  (fc) 

and 

F=1.60   VDh  W 

The  relation  between  the  diameters  of  rough  and  smooth 
pipes,  for  equivalent  discharges,  may  be  also  deduced,  giving  : 

Rough  Diameter 

---  =  1.15  (m) 

Smooth  Diameter 

That  is,  the  diameter  of  a  rough  pipe  to  give  the  same  dis- 
charge as  a  smooth  one  should  be  1.15  times  that  of  the  latter. 

It  is  to  be  remarked  that  if  q  =  U.  S.  gallons  per  minute, 
and  d  =  diameter  of  pipe  in  inches,  we  have  : 

c2 
-  =  o.8i  in) 

hd* 

This  is  an  intermediate  value  between  (d)  and  (/),  so  that  in 
a  great  many  cases  it  would  be  quite  safe  to  use  (d)  or  (/)  indif- 
ferently for  feet  and  seconds,  or  for  gallons,  inches  and  minutes. 

Comparison  with  observed  velocity  through  pipes  of  differ- 
ent diameters  and  different  degrees  of  roughness  and  smooth- 
ness* seems  to  establish  the  fact  that  the  above  series  of  formu- 
lae, from  (a)  to  (w),  both  inclusive,  cover  the  whole  range  of 
practical  cases,  and  that,  apart  from  entirely  abnormal  con. 
ditions,  no  cast-iron  pipe,  however  smooth,  will  give  a  greater 
discharge  than  that  deducible  from  (</)  and  (h),  nor  will  any  or- 

•  See  discussion  of  Mr.  Desmond  Fitzgerald's  paper,   "Flow  of  Water  in  48-in. 
Pipe,"  Vol.  XXXV.,  Transactions  American  Society  Civil  Engineers,  July,  1896. 


110  WATER  SUPPLY  ENGINEERING. 

dinary  degree  of  roughness  reduce  the  discharge  below  that  given 
by  (d)  and  (/). 

The  above  series  of  formulae,  therefore,  may  be  confidently 
used  in  all  practical  hydraulic  calculations,  to  the  exclusion  of 
all  others  given  in  the  first  part  of  this  book,  with  no  sacrifice  of 
accuracy,  and  a  great  gain  in  simplicity. 

It  must  be  again  repeated  that  since  the  tendency  of  all 
pipe  lines  is  to  become  more  or  less  incrusted  with  age,  and  as, 
besides,  many  other  causes,  such  as  leakage,  etc.,  tend  con- 
stantly to  diminish  the  quantity  discharged  by  any  given  pipe  line, 
it  is  no  more  than  common  prudence  to  adopt  always  the  formu- 
lae for  rough  pipes. 

Page  1+8.  —  As  an  example  of  the  use  of  the  table  on  this  page, 
let  it  be  required  to  know  the  diameter  of  pipe  necessary  to  dis- 

3 

charge  12  cu.  ft.  per  second,  with   a   fall   of  -       From    (d), 

we  have  : 


Consulting  the  table,  a  diameter  of  26  in.  is  found  to  corre- 
spond to  the  fifth  root  of  47.75.  Twenty-six  inches,  therefore,  is 
the  proper  diameter. 

All  that  precedes  has  had  reference  to  long  pipes,  when  only 
the  frictional  head  —  erroneously  so  called  —  has  been  considered. 
In  the  case  of  short  pipes,  the  exclusive  consideration  of  this  head 
is  no  longer  admissible.  The  method  to  be  then  pursued  will 
be  best  illustrated  by  an  example. 

Let  it  be  required  to  find  the  total  head  necessary  to  dis- 
charge 96  cu.  ft.  per  second  through  a  36-inch  pipe,  150  ft.  long. 
Calling  the  area  of  the  pipe  7  sq.  ft.,  the  required  Telocity  will 

96 
be  —  =  13.70  ft.  per  second.      The  head   necessary  to  produce 


WATER  SUPPLY   ENGINEERING.  Ill 

F« 

this  velocity,  according  to  the  laws  of  falling  bodies,  is .      We 

zg 

have  already  seen  (page  12)  that  the  head  necessary  to  overcome 
resistance  to  entrance  is  about  one-half  of  this,  so  the  two  com- 
3  F* 

bined  would  be .     The   frictional   head   per   1000  from  (t) 

4<7 
V*  15  F2 

is which  for  a  length  of  150  ft.  reduces  to -.   The 

1.61  x  3  483 

velocity  being  13.70,  F3  =  187.69.     Therefore  : 

3  F-      3  X  187.69 

Velocity  and  entrance  head  = = =  4.37  ft. 

40          4X32.2 

tf   T*  OF  THB  v 

15  X  187.69 

Frictional  head  = =  5.83  ft. 

483 
Total  head  above  center  of  pipe  10.20  ft. 

If  the  pipe  did  not  discharge  freely  in  air  (see  page  57)  there 
would  be  back  pressure  equal  to  entrance  head,  and  we  would 
have  : 

F' 

Velocity,  entrance  and  exit  head,  —  =  5.83  ft. 

0 
Frictional  head,  as  before  5.83  ft. 

Total  head,  above  center  of  pipe  11.66  ft. 

As  an  additional  example,  how  many  cubic  feet  per  second 
would  a  pipe,  24  in.  diameter  and  30  ft.  long,  discharge  freely 
in  the  air  from  a  reservoir,  its  center  being  12  ft.  below  the  sur- 
face of  the  water  ? 

The  velocity  and  entrance  head,  using  round  numbers,   is 

3  F3      F2  F3 

=  — .      The  frictional   head   per  1000  from    (i)   is  . 

4*7        43  3.22 

3F2 

The  friction   head   for  the  given  length,  30  ft.,  will  be . 


112  WATER  SUPPLY  ENGINEERING. 

The  total  head  above  center  of  pipe  being  12  ft,  we  have  : 

F2        3  F2  /I  3    \ 

12  =   ---  1  ---  =    F2  (  --  1  --  ) 

43         322  V  43         322  / 

Using  round  numbers  : 

/I  1  \  7107  +  43\ 

12=  F2    (-  +  -  )   =  F2     --  ) 
M3         107/  \      4600    / 

F  =  19ft.  per  second. 

The  quantity  discharged  being  equal  to  the  velocity  multi- 
plied by  the  area  of  the  2  ft.  pipe  is  19  -x  3.14  =  59.66  cu.  ft. 
per  second. 

It  will  be  observed  that  "round  numbers"  have  been  freely 
used  in  the  above  calculation.  A  great  deal  of  unnecessary  fig- 
uring can  be  avoided,  and  an  ample  degree  of  accuracy  secured, 
by  a  judicious  discarding  of  small  decimals  in  all  hydraulic 
work. 

Page  70.  —  Many  other  useful  data  can  be  added  to  the  few 
already  given.  Thus: 

%  Cu.  ft.  per  sec.   X    86400     =  en.  ft.  per  24  hours 

X  646272      =  U.  S.  gallons  per  24  hours. 
X    26928     =       "  "    hour. 

X       448  8  =  "    minutes 


93.00  CU'"ft'  "emfn:  }  =  10017-2  U'  S'  ^allon8  Per  24  hour8' 

It  is  therefore  convenient  to  remember  that  1.5  cu.  ft.  per  sec- 
ond equals  very  closely  1,000,000  gallons  per  24  hours. 
Also  : 

1  acre  covered  1  in.  =    3630  cu.  ft.  =    27,152  U.  S.  gallons. 

1    "  "1  ft.  =  43560     "        =  325,829      " 

1  square  mile  covered  1  in.  =    2323200  cu.  ft.  =     17377536  U.  S.  gallons. 

1        ......         1  ft.  =  27878400      "        =   208530432     " 

Also  : 


4000 

H  P  =  net  or  theoretical  horse-power. 
G  =  U.  S.  gallons  per  minute. 
H  —  height  in  feet  to  which  water  is  raised. 

Also,  for  discharge  over  weirs  or  spillways  : 

Q  =  2000000  L  Vtf*. 

Q  =  U.  S.  gallons  per  24  hours  (close  approximation). 

L  =  length  of  weir  in  feet. 

H  =  height  from  sill  of  weir  to  surface  of  still  water,  in  feet. 


WATER  SUPPLY   ENGINEERING.  J13 

The  horse-power,  HP,  of  water  falling  over  a  weir  : 

H  P  =  0.35  L  X  F    VH*. 

F  -  fall  in  feet,  other  symbols  as  above. 

Also,  for  weight  in  long  tons  of  cast-iron  pipe,  per  mile,  we 
have  : 

P  =  0  20  wM. 

P  =  25  M  (D  +  T)  T. 

P  —  approximate  weight  in  long  tons  (2,240  Ibs.)  of  pipe  line. 

w  =  weight  in  Ibs.  of  12  ft.  lengih  over  all  of  pipe. 

M  —  length  of  pipe  line  in  miles . 

D  =  inside  diameter  of  pipe  in  inches. 

T  —  thickness  of  pipe  in  inches. 

All  the  formulas  given  in  this  book  refer  to  cast-iron  pipe. 
We  are  as  yet  without  sufficient  experimental  data  as  to  how  far 
they  may  be  applicable  to  wrought-iron  or  steel-riveted  pipe.  It 
appears,  however,  that  such  pipe  do  not  give  as  high  velocities 
as  cast-iron  pipe  of  the  same  diameter.  It  is  probable  that  the 
diameters  of  wrought-iron  or  steel  lap-jointed  riveted  pipe  should 
be  from  5  per  cent,  to  10  per  cent,  greater  than  those  of  cast- 
iron  pipe,  to  insure  an  equal  discharge. 

Page  71. — Chemical  science  has  not  yet  reached  the  point 
of  drawing  definite  conclusions  from  an  analysis  of  water.  Still 
a  chemical  examination  should  always  form  part  of  the  investi- 
gation of  the  quality  of  any  proposed  water  supply,  as  it  is  fre- 
quently of  great  utility,  when  combined  with  other  information, 
in  enabling  an  intelligent  judgment  to  be  formed  of  the  whole- 
someness  of  the  supply. 

The  element  of  danger  most  to  be  dreaded  in  a  water  supply 
is  sewage  contamination,  and  it  is  to  the  detection  of  the  evi- 
dences of  such  contamination  that  the  efforts  of  the  chemist  are 
mainly  directed  in  an  examination  of  a  water  sample.  The  sub- 
stances regarded  with  the  greatest  suspicion  are  albuminoid 
ammonia,  chlorine,  and  nitrites.  In  regard  to  these,  the  follow- 
ing quotations  from  Mr.  C.  C.  Vermeule's  Report,  forming  Vol. 
III.  of  Final  Report  of  the  State  Geologist  of  New  Jersey,  will 
be  found  very  useful: 


114  WATER  SUPPLY  ENGINEERING. 

"Albuminoid  Ammonia. — This  represents  animal  and  veget- 
able matter  present  in  the  waters  and  in  process  of  decomposition, 
by  which  process  free  ammonia  is  produced,  consequently  it  is 
more  to  be  dreaded  than  the  latter,  as  at  certain  stages  of  de- 
composition such  matter  becomes  very  dangerous  to  health.  Dr. 
Leeds'  limit  is  0.028  (parts  in  100,000). 

"  Chlorine. — This  is  not  only  an  accompaniment  of  sewage 
pollution,  but  is  a  measure  of  the  amount  of  such  pollution,  al- 
though not  always  of  the  danger  to  be  apprehended  therefrom. 
Dr.  Leeds  gives  the  maximum  allowable  at  one  part  in  100,000. 

"Nitrates  and  Nitrites. — Pollution  by  sewage  being  practi- 
cally the  addition  of  nitrogen  compounds  to  the  water,  the  process 
of  purification  of  this  water  consists  of  the  oxidization  of  these 
compounds,  and  when  this  process  is  completed  they  become 
nitrates.  Nitrites  indicate  that  this  work  of  purification  is  in 
progress,  but  is  not  complete,  consequently  their  presence  is  a 
more  serious  matter  than  that  of  nitrates." 

Mr.  J.  T.  Fanning,  in  his  valuable  work  on  water-supply 
engineering,  gives  as  a  quotation  Heisch's  "Simple  Sugar  Test 
of  Water,"  as  follows  : 

"If  half  a  pint  of  the  water  be  placed  in  a  clean,  colorless- 
glass  stoppered  bottle,  a  few  grains  of  the  best  white  lump  sugar 
be  added,  and  the  bottle  freely  exposed  to  the  daylight  in  the 
window  of  a  warm  room,  the  liquid  should  not  become  turbid,  even 
after  exposure  for  a  week  or  ten  days.  If  the  water  becomes 
turbid,  it  is  open  to  grave  suspicion  of  sewage  contamination  ; 
but  if  it  remain  clear,  it  is  almost  certainly  safe." 

Page  74. — Within  the  last  few  years,  indeed  since  the  appear- 
ance of  the  first  edition  of  this  little  volume,  two  features  of  water- 
supply  engineering  hitherto  but  little  considered  in  this  country 
have  forced  themselves  into  notice.  These  are  artesian,  or 
driven  wells,  and  the  filtration  of  public  water  supplies. 

.  Artesian  wells  have,  it  is  true,  been  long  used  for  small  sup- 


WATER  SUPPLY  ENGINEERING.  115 

plies,  but  of  late  years  their  use  on  a  large  scale  has  taken  a 
great  development.  In  many  places  they  afford  the  only  means 
of  supply,  and  after  a  vain  attempt  to  secure  surface  water  in 
sufficient  quantities  and  of  a  proper  quality,  it  becomes  frequently 
necessary  to  have  recourse  to  the  vast  supplies  stored  away  be- 
neath the  surface,  and  which  may  be  exploited  or  mined  like 
any  other  subterranean  deposit  of  valuable  material. 

The  employment  of  these  wells  has  natural  limitations. 
They  require  the  existence  of  a  suitable  geological  formation, 
without  which  they  would  be  unproductive.  They  must  reach 
a  permeable,  deep-seated  stratification,  which  is  found  onlv,  and 
by  no  means  always,  in  the  secondary  and  tertiary  formations. 
Of  these,  the  supply  taken  from  the  secondary  is,  while  of  greater 
rarity,  generally  of  more  considerable  volume  than  that  furnished 
by  the  tertiary. 

In  some  of  these  artesian  wells  the  water  reaches  to  the  sur- 
face of  the  ground,  and  even  spouts  out  to  a  considerable  height 
above  it.  This  greatly  facilitates  the  practical  utilization  of  the 
supply.  In  others  the  water  reaches  only  to  a  considerable  depth 
below  the  surface,  when  the  difficulties  of  raising  and  distributing 
it  become  greatly  increased.  If  the  whole  supply  is  furnished  by 
a  single  boring  the  problem  is  much  simplified,  because  the  pump 
can  be  placed  directly  over  the  well  and  a  lifting  main  used  to 
bring  the  water  within  reach  of  the  force  main,  without  the  em- 
ployment of  a  separate  pump. 

If,  however,  as  is  generally  the  case,  a  gang  of  wells  is  neces- 
sary^ the  expense  of  getting  the  water  to  the  force  main  is  greatly 
increased,  unless  the  water  in  the  wells  rises  to  within,  say,  20  ft. 
of  the  surface.  When  this  occurs,  the  yield  of  the  whole  gang 
can  be  collected  in  a  single  suction  main,  feeding  to  the  pump. 
Otherwise  each  well  must  have  a  separate  pump.  It  is  probable 
that  in  such  oases  power  might  be  advantageously  transmitted 
electrically  from  a  single  engine  to  all  the  auxiliary  pumps,  or 


116  WATER  SUPPLY  ENGINEERING. 

some  system  of  air  lifts  might  be  used,  operated  by  a  single  com- 
pressor. 

In  studying  the  project  of  a  driven-well  system  for  any  par- 
ticular locality,  although  general  inferences  may  be  drawn  from 
the  geology  and  topography  of  the  district,  no  positive  knowledge 
can  be  obtained  without  sinking  experimental  wells.  Without 
such  practical  tests,  all  forecasts  are  but  little  better  than  guess- 
work. 

Page  76. — In  estimating  the  amount  of  water  required  for 
any  locality,  the  quantity  needed  for  fire  service  must  be  carefully 
considered.  For  small  towns,  it  will  be  found  that  if  provision  is 
made,  both  as  to  quantity  required  per  minute,  and  pipe  capacity 
to  convey  the  same,  in  regard  to  fire  service,  it  will  more  than? 
cover  all  other  needs,  while  in  the  case  of  a  large  city  the  water 
needed  for  fire  service  is  a  relatively  small  proportion  of  the 
whole,  and  if  all  the  requirements  of  domestic  and  the  other 
public  needs  are  met,  no  further  provision  is  necessary  for  fire- 
service. 

According  to  Mr,  Fanning,  a  4-in.  pipe  is  required  to  supply 
a  single  hose  stream  of  about  20  cu.  ft.  per  minute.  A  6-in. 
pipe,  according  to  the  same  authority,  will  furnish  two  such 
streams.  It  is  quite  clear,  therefore,  that  for  the  smallest  town 
a  4-in.  main  is  the  absolute  minimum,  while  for  towns  of  any 
considerable  size,  either  actual  or  prospective,  6  ins.  is  the  small- 
est sized  pipe  that  should  be  laid  in  any  of  its  streets.  As  a 
general  rule,  it  may  be  stated  that  each  street  main  should  be 
sufficient  to  furnish  enough  water  for  at  least  one  ordinary  con- 
flagration in  that  street. 

Page  77. — All  extensive  exposed  water  surfaces,  such  as  lakes 
and  large  ponds,  lying  within  the  survey,  should  be  excluded 
from  the  estimate  of  available  watershed  area,  because  the  evapo- 
ration from  such  surfaces  is  practically,  equal  to  the  rainfall  upon 
them. 


WATER  SUPPLY  ENGINEERING.  117 

Page  80. — A  convenient  formula,  and  one  which  it  is  be- 
lieved will  prove  in  most  cases  to  be  approximately  correct,  for 
the  necessary  amount  of  storage  is  the  following : 

c» 
s=- 

Y 

In  this  formula,  S  =  the  amount  of  storage  required,  C  — 
the  yearly  consumption,  and  Y  —  the  yearly  available  yield  of 
the  watershed,  all  expressed  in  the  same  unit. 

Example. — A  town  consumes  10  million  gallons  per  day,  or 
3,650  millions  per  year.  The  estimated  yearly  yield  of  the  source 
whence  the  supply  is  drawn  is  ten  times  that  amount,  or  36,500 
millions.  What  storage  capacity  is  needed  to  insure  the  required 
daily  amount  throughout  the  year  ? 

13322600 

8  = =  385  million  gallons. 

36500 

This  is  equivalent  to  storage  of  36.5  days'  supply,  or  that  of 
about  five  weeks. 

Page  S3.— In  the  calculations  given  on  this  and  subsequent 
pages,  no  account  has  been  taken  of  the  resistance  to  slid  ing  upon 
the  base.  It  is  not  generally  necessary  to  do  so  in  the  case  of 
masonry  dams,  because  such  structures  always  are,  or  should  be, 
built  upon  a  rock  foundation,  and  therefore  cannot  be  pushed 
forward  without  shearing  through  the  joints  of  the  masonry.  If 
a  wall  rests  on  a  slippery  foundation,  however,  there  may  be  great 
danger  of  its  sliding  forward  long  before  it  can  yield  by  over- 
turning. The  resistance  which  a  wall  or  dam  sustaining  water 
pressure  offers  to  sliding  is  its  weight  multiplied  by  some  *f  co- 
efficient of  friction/'  always  taken  as  less  than  unity,  while  the 
force  tending  to  produce  sliding  is  the  thrust  of  the  water  =  31.25 
Hz,  H  being  the  depth  of  water  pressing  against  the  wall. 

If  we  assume  a  density  of  125  Ibs.  per  cubic  foot  as  that  of 
the  wall,  and  represent  its  thickness  by  By  assuming  also  a  co- 


118  WATER  SUPPLY   ENGINEERING. 

efficient  of  friction  of  0.75,  then  if  a  plumb  wail  sustains  water 
pressure  against  its  full  height  //.  the  equation  of  equilibrium  is: 

31.25  H*  =  0.75  X  125  X  H  X  B. 

H 
B  =  — 

A  thickness  equal  to  two-thirds  the  height  of  such  a  wall 
would  therefore  give  a  factor  of  safety  of  2,  as  against  sliding,  but 
this  factor  would  be  reduced  if  the  wall  were  transformed  to  a 
trapezoidal  section  according  to  Vauban/s  rule,  because  the 
weight  would  be  diminished. 

Two  excellent  formulae,  readily  deducible  from  the  princi- 
ples already  laid  down,  may  be  used  for  determining  the  bottom 
width,  b,  of  a  trapezoidal  wall  sustaining  water  pressure  to  its 
full  heigh th,  h,  when  its  top  width,  t,  and  the  density,  d.  of  the 
material  of  which  it  is  composed  are  given,  as  well  as  the  desired 
factor  of  safety,/,  and  the  coefficient  of  friction,  c.  They  are, 
as  against  sliding  : 

62. 50  hf 


cd 


and  as  against  overturning 


Example.— Let  h  =  30  ft.,  t  =  6  ft.,  d  =  125  Ibs.,   /  =  2, 
and  c  —  0.75.     Then,  as  against  sliding  : 


62.50  X  30  X  2 

6  = 6  =  34  ft. 

0  75  X  125 


As  against  overturning  : 


.        ,  .      /125  X  2  X  900  6 

b  =  X,i/ +  3  X  56 =  18.85  ft. 

125  2 

These  results  bring  out  in  bold  relief  what  has  been  already 
said  regarding  possible  danger  of  sliding  when  the  wall  is  not 
well  tied  down  to  a  solid  foundation. 


WATER  SUPPLY  ENGINEERING. 


119 


A  few  memoranda  regarding  hydrostatic  pressure  against 
wet  surfaces  will  be  useful  in  this  connection. 

As  regards  pressures  against  plane  vertical  surfaces,  nothing 
need  be  added  to  what  has  already  been  said  in  reference  to  dams 
(page  83,  etc.),  but  some  additional  information  regarding  in- 
clined and  curved  surfaces  will  be  useful. 

Pressure  upon  the  plane  inclined  surface,  of  which  the  edge 
is  shown  by  the  line  A  B  (fig.  22),  and  of  which  the  length,  at 


FIG.  22. 

right  angles  to  the  plane  of  the  paper,  is  supposed  for  conve- 
nience to  be  1  foot,  is  of  three  kinds.  First  we  have  total  pres- 
sure in  pounds  which  is  equal  to  the  length  of  the  line  A  B  mul- 
tiplied bv  half  the  height  A  C,  and  by  62.5.  The  point  of  ap- 

A  B 

plication  of  this  pressure  is  at  the  distance from  A,    and  is 

3 
exercised  in  a  direction  normal  to  A  B. 

This  total  pressure  is  divided  at  the  point  of  application 
into  two  components,  which  form  the  two  other  pressures  re- 
ferred to,  one  vertical  and  downward,  and  the  other  horizontal 
and  outward.  The  first  of  these  is  the  total  vertical  downward 
pressure  upon  the  surface  represented  by  A  B,  and  is  equal  to 


120 


WATER  SUPPLY  ENGINEERING. 


the  weight  of  water  resting  upon  it,  or  the  area  of  triangle   A  B  C, 
multiplied  by  62.5  Ibs.,  and  the  other  is  the  horizontal  thrust 


AC  _8 

and  is  equal  to  multiplied  by  62.5  Ibs.  =  31  25   4  C- 

2 


overturning    moment    of    this   latter  pressure    is 
3  ft.  Ibs. 


62.5^4   C 


10.42  A  0 

The   pressures   upon   the  curved   surface  A  B  (Fig.  23)  are 
naturally  more  complicated.     The  total  pressure  is  best  deter- 


FlQ.  23* 

mined  graphically  by  dividing  A  B  into  a  certain  number  of  short, 
equal  straight  lines  A  b,  b  c,  etc.,  thus  converting  the  curve  into 
a  nearly  equivalent  polygon.  The  total  pressure  upon  these  ele- 
ments will  be  A  b  x  g\  b  c  x  g',  etc.,  g,  g' ,  g",  etc.,  being  the 
vertical  distances  from  the  middle  of  the  lines  A  b,  b  c,  c  d,  etc., 
to  the  surface  of  the  water,  and  the  total  pressure  upon  A  B  will 
be  given  by  their  sum.  The  total  vertical  and  horizontal  pressures 
will  be  the  sums  of  those  of  the  elements  A  b,  b  c,  etc.,  and  will 


WATER  SUPPLY   ENGINEERING.  121 

foe,  therefore,  the  one  equal  to  the  weight  of  the  volume  of  water 

i 

A  ~G 
ABC,  and  the  other  equal  to  -          the  same  as  for  the  plane  in- 

'/i 

dined  surface. 

The  point  of  application  of  this  horizontal   thrust  is  at  the 
A   C 

height  from  A.     It  will  be  observed  that  both   the  thrust 

3 

and  its  point  of  application  are  the  same  for  vertical  and  inclined 
plane  surfaces,  and  also  for  curved  surfaces. 

The  point  of  application  of  the  vertical  downward  pressure, 
both  for  plane  inclined  and  curved  surfaces,  passes  through  the 
centre  of  gravity  of  the  volume  of  water  resting  upon  the  sur- 
face. 

Page  87. — In  the  author's  opinion  the  two  formulae  (9)  and 
(10)  may  be  replaced  by  the  single  general  one  : 

/  L-D  \ 

p  =  I 1  w. 

\L.D) 

This  formula  agrees  with  both  of  the  others  for  D  —  _,  and 

3 

with   (10)  for  D  =  _.     It  also  agrees  very  closely  with  (10)  for 

2 
all  intermediate  values  of  D.     It  gives  values  increasingly  greater 

than   (9)  for  all  values  of  D  between    —  and  o,  which  in  the 

3 

author's  opinion  it  should  do,  since  we  are  warned  by  good 
authorities  that  much  confidence  is  not  to  be  placed  in  (9)  when 

D  becomes  more  than  slightlv  less  than  J.     The  author  is  also 

3 

inclined  to  the  belief  that  the  pressure  is  distributed  more  or  less 
uniformly  throughout  the  entire  length  of  D,  and  not  concen- 
trated at  its  extremity. 


122  WATER  SUPPLY   ENGINEERING. 

Page  92. — In  regard  to  the  section  shown  in  Fig.  19,  it  would 
be  improved  by  making  the  slope  D  B  a  little  steeper,  and  the 
slope  C  E  a  little  flatter,  than  shown,  so  as  to  increase  the  unit 
stress  at  D  and  reduce  it  at  E.  It  is  safer,  also,  in  these  calcula- 
tions to  discard  the  moment  of  the  water  pressing  vertically  down- 
ward against  the  inclined  inner  side  of  the  dam,  as  its  action 
through  the  mass  of  masonry  is  somewhat  uncertain.  It  will 
always  be  an  additional  factor  of  safety,  but  it  is  doubtful  if  its 
amount  can  be  accurately  determined  as  to  its  practical  action. 

Pages  98-99. — In  regard  to  the  manner  of  drawing  water 
from  a  reservoir,  sliding  sluice  gates  have  been  of  late  so  per- 
fected in  their  design,  and  have  proved  so  efficient  and  reliable 
in  practice,  that  one  of  the  valves  recommended  in  the  text  may 
be  advantageously  replaced  by  a  sluice  gate  of  approved  make, 
set  against  the  mouth  of  the  pipe,  inside  of  the  tower. 


THIRD  PART. 

FLOW   OF  WATER  THROUGH  MASONRY  CONDUITS,  AND    SOME 
DETAILS  OF  TUNNEL  CONSTRUCTION. 

From  his  experimental  researches,  Darcy  deduced  formulae 
for  the  mean  velocity  of  water  flowing  through  open  canals,  or 
any  other  form  of  conduit  laid  to  a  uniform  grade,  and  not  run- 
ning under  pressure.  The  formulae  vary  with  the  degree  of 
smoothness  of  the  interior  surface  of  the  conduits.  The  formula 
which  he  established  for  smooth,  though  not  polished,  surfaces, 
and  which  is  the  one  suited  most  nearly  to  well-constructed, 
brick-lined  aqueducts,  is  as  follows: 

R  1  /         0.07  \ 

-    =     0.00019  -- 


/  0.07 
(  H  -- 
\  R 


In  which  R  —  hydraulic  mean  radius,  or  the  water  section 
divided  by  the  "wet  perimeter";  /=  the  slope  per  unit  of 
length,  or  the  total  fall  divided  by  the  total  length,  and  U  the 
mean  velocity  of  the  current.  This  formula  applies  to  the  metric 
system,  and  £7  is  expressed  in  meters  per  second. 

To  adapt  the  above  to  feet,  it  may  be  most  conveniently 
written  as  follows  : 


R I  /  0.46  \ 

=   0.00001    (6.6H )  0) 

V  \  R  ) 


Whence 


6.6  R  4-  0.46 

In  which  U  =  mean  velocity,  in  feet  per  second. 


134  WATER  SUPPLY  ENGINEERING. 

As  a  simple  illustration  of  the  use  of  this  formula,  suppose  a 
canal,  6  ft.  wide,  with  vertical  sides,  laid  to  a  slope  of  ^-^  = 
0.001  and  running  3  ft.  deep.  The  water  section  is  then  18  sq. 
ft.,  the  wet  perimeter  12  ft.,  and  the  mean  hydraulic  radius  is 
1.5. 

Substituting  the  above  data  in  (2)  : 


100000  X  0.001 
6.6X1.5  +  0.46    =4-67  ft.  per  8eo. 

It  is  to  be  noted  that  the  above  formulas  (1)  and  (2)  apply 
only  to  conduits  of  sufficient  length  and  uniformity  to  permit 
of  the  establishment  of  what  is  called  the  "permanent  regimen." 
This  occurs  for  any  canal  or  channel,  when  the  depth  of  water 
is  uniform  throughout  its  entire  length;  that  is,  when  the  surface 
of  the  water  is  parallel  to  the  bottom  of  the  conduit. 

In  the  special  case  of  a  circular  section,  running  full  but  not 

under  pressure,  R  —  -j-,  D  being  the  diameter  in  feet.     It  will 

be  interesting  to  see  how  the  velocity  through  a  brick-lined  con- 
duit 4  ft.  in  diameter,  as  given  by  the  above  formula,  compares 
with  that  of  a  smooth  cast-iron  pipe  of  the  same  diameter.  As- 
suming a  fall  of  1{*00  in  both  cases,  we  have  for  the  brick 
conduit : 


=     /iooooo2Lo.ooi 

r        R  R  4 .  ft  ifi 


6.6  +  0.46 

For  a  smooth  cast-iron  pipe  of  same  diameter,  we  have  from 

*  (*)•• 

V  =  1.80  V4  =  3.60  ft.  per  second 

This  is  slightly  less  than  that  through  the  brick-lined  con- 
duit. The  increased  velocity  through  such  conduits  seems  to  be 
due  to  the  fact  that  they  are  laid  to  a  true  uniform  grade,  with 
few  changes  of  direction,  and  present  a  uniform  and  unbroken 
interior  surface. 


WATER  SUPPLY  ENGINEERING.  125 

For  coDduits   lined   with   good    rubble  masonry,  the  mean 
velocity  V  may  be  taken  as  : 


TT'=TTJ   19* +  1.33 
r        24  R  -4-  60  ' 


24  R  +  60 

U  being  the  mean  velocity  through  a  brick-lined  conduit  of  the 
same  elements.  U'  will  generally  be  from  80$  to  90$  of  £7. 

The  application  of  a  formula  to  such  a  conduit  is  very  un- 
certain, because  the  projections  of  the  rubble  masonry  render  it 
impossible  to  determine  the  exact  area  of  waterway. 

The  maximum  discharge  through  a  circular  or  a  "  horse- 
shoe" shaped  conduit  does  not  occur  when  running  entirely  full 
(unless  it  is  under  pressure),  but  when  full  to  within  about  ^th  of 
the  radius  of  the  crown,  because  the  velocity  and  the  wet  section 
are  then  in  the  most  favorable  relation  to  each  other.  Thus,  the 
discharge  through  a  circular  conduit  6  ft.  in  diameter  would  be 
greatest  when  it  was  running  full  to  within  0.30  ft.  of  the  top. 
In  a  circular  conduit  the  velocity,  when  running  half  full,  is  pre- 
cisely equal  to  that  when  running  entirely  full,  so  the  discharge 
is  just  one-half.  It  is  said  that  the  most  favorable  section  of  the 
horseshoe  type  is  when  the  total  height  is  equal  to  the  greatest 
width,  which  will  naturally  be  at  the  spring  line  of  the  arch.  This 
assimilates  the  section  to  a  circle. 

As  to  the  best  form  of  cross-section  for  masonry  conduits  of 
large  dimensions,  the  circle  has  much  to  recommend  it  as  regards 
strength,  economy,  and  delivery.  It  can  be  readily  lined  with 
bricks  of  ordinary  shape  and  dimensions.  On  the  other  hand, 
the  horseshoe  form  affords  more  floor  room,  and  possesses  certain 
facilities  for  construction,  among  others  that  of  permitting  the 
building  of  the  sides  and  arch  of  the  lining  before  the  invert  is 
put  in,  as  was  very  successfully  done  in  building  the  new  Croton 
aqueduct.  The  driving  of  a  tunnel,  when  the  excavation  and 
masonry  lining  are  carried  on  simultaneously,  is  an  exceedingly 
troublesome  piece  of  work,  on  account  of  the  confined  space  in 


126  WATER  SUPPLY   ENGINEERING. 

which  the  operations  are  executed  and  the  great  amount  of  mate- 
rials passing  in  and  out.  If  the  invert  is  built  first,  as  would  seem 
to  be  the  natural  order  of  executing  the  work,  it  is  exposed  to 
injury,  and  greatly  impedes  drainage  and  also  the  hauling  of 
materials. 

For  these  and  other  reasons  it  is  a  great  advantage  to  leave 
the  invert  till  all  the  rest  of  the  work  is  completed  from  shaft  to 
shaft.  The  horseshoe  form  permits  of  this  very  well,  but  the 
use  of  special  bricks  is  required  to  make  the  junction  between 
the  side  walls  and  the  invert.  Such  special  bricks  were  used  in 
the  new  Croton  aqueduct.  A  course  of  cut  stone  blocks  or  skew- 
backs  would  be  far  better  than  the  special  bricks,  but  very  much 
more  expensive. 

One  of  the  most  important  details  of  tunnel  construction  is 
the  backing  behind  the  side  walls  and  over  the  arch.  Nothing 
is  really  satisfactory  but  wet  masonry  of  some  sort,  although  for 
economy  dry  work  is  sometimes,  perhaps  generally,  specified. 
The  effect  of  dry  stone  packing  over  an  arch  is  really  little  more 
than  loading  it  very  irregularly  with  a  heap  of  loose  stones  and 
small  material.  At  all  events  the  spandril  backing  should  be 
carried  up  to  the  level  of  the  extrados  of  the  crown  of  the  arch 
in  cement  masonry.  All  small  vacancies  are  probably  best  taken 
up  with  wooden  lagging  packed  tight.  In  a  tunnel  this  lagging 
lasts  a  long  time  and  affords  an  elastic  cushion  to  deaden  the 
shock  of  any  movement  of  the  roof  which  may  occur.  By  the 
time  it  decays,  if  it  ever  does  decay,  the  roof  will  have  adapted 
itself  to  its  final  bearings,  and  the  cement  of  the  masonry  lining 
and  walls  will  have  become  thoroughly  indurated. 

The  alignment  of  the  new  Croton  aqueduct  demonstrated 
the  surprising  accuracy  with  which  the  centre  line  of  a  tunnel 
can  be  ranged  between  plumb  lines  dropped  from  the  two  sides 
of  a  shaft.  As  a  useful  detail,  it  may  be  interesting  to  state  that 
it  was  found  that,  owing  to  its  optical  properties,  a  telescope, 


WATER  SUPPLY   ENGINEERING.  127 

placed  inline  with  two  plumb  lines,  with  the  vertical  cross  hair 
exactly  covering  both,  could,  if  placed  sufficiently  near  the 
nearest  one,  focus  it  entirely  out  of  sight,  the  visual  rays  passing 
around  it,  on  both  sides,  and  focusing  on  the  farther  line.  This 
property  was  exceedingly  useful  insetting  the  instrument  in  line. 
Sewers  are  often  built  with  an  egg-shaped  cross-section  the 
point  downward.  This  form  seems  best  suited  to  accommodate 
a  variable  flow,  although  the  circle  has  much  to  recommend  it 
for  sewers  also. 

FILTRATION. 

The  development  of  the  nitration  of  public  water  supplies  is 
a  recent  feature  of  hydraulic  engineering  in  this  country. 
Hitherto  this  process  has  been  looked  upon  with  general  disfavor 
and,  in  the  case  of  new  water-works,  the  endeavor  has  always 
been  to  secure  an  unobjectionable  supply  of  natural  water.  This 
is  perfectly  proper,  when  practicable,  but  year  by  year  the  at- 
tempt becomes  more  difficult,  and  besides,  supplies  which  when 
first  used  were  perfectly  wholesome  may  become  so  polluted  by 
gradual  encroachments  as  to  be  rendered  unfit  for  use.  In  this 
manner  attention  is  now  being  turned,  in  the  United  States,  to 
the  practical  application  of  those  processes  of  purification  which 
have  been  so  long  and,  as  it  seems,  successfully,  employed  in 
Europe. 

Another  reason  for  the  growing  favor  with  which  filtration 
on  the  large  scale  is  now  being  regarded  is  the  fact,  recently  es- 
tablished, that,  contrary  to  previous  belief,  filtration  is  not  con- 
fined in  its  action  to  a  mere  retention  of  matters  held  in  suspen- 
sion, but  also  exercises  a  marked  chemical  effect  upon  impure 
waters. 

Although  our  knowledge  of  the  true  action  of  filtration  has 
greatly  advanced,  it  is  still  far  from  being  complete,  and  the 
precise  process  by  which  the  chemical  purification  is  effected  is 
still  under  discussion.  It  seems  however  certain  that  the  sedi- 


128  WATER  SUPPLY  ENGINEERING. 

mentation  layer,  or  the  gelatinous  film  formed  by  deposition- 
from  the  water  itself,  plays  a  great  part  in  that  destruction  of  a 
large  percentage  of  noxious  bacteria  which  is  found  to  be  the 
result  of  a  proper  system  of  filtration. 

Owing  to  the  great  areas  required  for  filter  beds — about  one 
acre  per  day,  per  24  million  gallons,  "  mechanical  filters  "  (so- 
called)  are  largely  used  in  the  United  States.  These  filters  are 
much  more  rapid  in  their  action  than  the  ordinary  filter  beds, 
with  the  necessary  result  of  being  less  efficacious.  In  order  to 
increase  their  efficiency,  a  coagulant — generally  alum — is  often 
used.  The  sulphate  of  alumina  precipitates  rapidly  many  of  the 
impurities  of  water,  and  then  disappears  with  them  wholly  or 
nearly  from  the  effluent. 

The  following  particulars  are  given  by  the  manufacturers  of 
a  leading  mechanical  filter  :  A  fixed  solution  of  sulphate  of 
alumina  is  used,  one  pound  sufficing  for  the  clarification  of  from 
50,000  to  150,000  Ibs.  of  water.  The  filtering  material  is  crushed 
crystal  white  quartz,  "assisted,  after  the  introduction  of  the  sul- 
phate of  alumina,  by  the  gelatinous  mineral  film  of  hydroxide  of 
alumina.  This  sensitive  membrane,  with  its  fine  quartz  founda- 
tion, is  so  powerful  as  to  be  capable  of  retaining  all  coloring 
matter  and  bacteria."  For  city  supply,  the  unit  filter  passes 
ordinarily  250  gallons  per  minute.  When  clean,  one  foot  of  head 
passes  this  quantity.  As  the  filter  silts  up  the  necessary  head 
increases  progressively.  When  it  reaches  8  ft.  cleaning  be- 
comes necessary.  Cleaning  is  effected  by  passing  a  sufficient 
quantity  of  water  through  the  crushed  quartz  while  it  is  being 
thoroughly  stirred  up  by  a  special  appliance. 

"  Initial  cost  of  plant  exclusive  of  land,  foundations  and 
buildings,  depends  on  valocity  of  filtration  ;  it  will  average,  deliv- 
ered, erected  and  connected,  $6,000  per  million  gallons.  When  the 
plant  is  cared  for  by  an  engineer  required  for  other  purposes  as 
well,  the  cost  of  maintenance  is  about  $2. 50  per  million  gallons/' 


WATER  SUPPLY  ENGINEERING.  129 

On  the  subject  of  filtration  consult  Hazen's  "  Filtration  of 
Public  Water-Supplies"  and  Mason's  "Water-Supply." 

PUMPS  AND  PUMPING  ENGINES. 

In  many  instances  the  source  of  a  water  supply  lies  lower 
than  the  locality  to  which  it  is  to  be  delivered.  In  such  cases 
pumping  must  be  resorted  to. 

A  pumping  plant  consists  essentially  of  a  suction  and  force 
main,  with  a  pump  working  between  them. 

It  is  clear  that  the  suction  valves  of  the  pumps  must  always 
be  sufficiently  close  to  the  level  of  the  supply  to  ensure  a  steady 
and  powerful  draft  at  the  suction  end.  This  necessitates  the 
placing  of  the  whole  plant  at  a  low  level,  and  when  the  elevation 
of  the  water  in  the  sump  is  subject  to  fluctuations,  as  in  the 
case  of  its  being  fed  from  a  river,  which  may  rise  and  fall  con- 
siderably according  to  the  season,  the  action  of  the  pumps  may 
be  seriously  impeded.  The  old  Cornish  type  of  pumping  engine 
is  comparatively  free  from  this  difficulty,  because  the  plant  can 
be  located  at  any  distance  above  the  suction  chamber,  but  for 
many  reasons  this  type  of  .engine  is  not  now  employed  in  this 
country  for  pumping  large  water  supplies. 

Pumping  engines  may  be  divided  into  the  three  classes  of  low, 
medium  and  high  duty.  This  classification  refers  entirely  to  the 
relative  consumption  of  fuel  to  accomplish  a  given  delivery  of 
water.  Broadly  speaking,  a  low  duty  engine  is  one  consuming 
more  than  4  Ibs.  coal  ;  a  medium  duty  engine,  one  that  consumes 
between  2  and  4  Ibs.,  and  a  high  duty  engine,  one  that  works  with 
2  Ibs.  or  less  of  coal,  per  hour  per  horse  power.  It  is  evident  that 
the  low  duty  type  includes  all  single  cylinder  engines,  that  the 
medium  duty  type  calls  for  a  compound  engine,  while  the  third 
or  high  duty  class  demands  the  most  refined  type  of  triple  expan- 
sion engine  using  an  automatic  cut-off,  and  all  other  heat  saving 
appliances. 

The  selection  of  one  of  +hese  types  in  any  particular  case  re- 


130  WATER  SUPPLY  ENGINEERISG. 

quires  a  great  deal  of  judgment,  and  depends  upon  the  required 
daily  supply,  the  cost  of  fuel,  facilities  for  repairs,  the  nature  of 
the  help,  as  well  as  the  first  cost  of  the  plant.  A  high  duty 
pumping  engine  is  not  only  very  expensive  as  to  first  cost,  but  it 
is,  of  necessity,  a  complicated  piece  of  mechanism,  requiring  the 
best  skill  for  its  care  and  operation,  and  exceptional  facilities  for 
repairs  in  case  of  a  breakdown.  It  must  be  borne  in  mind,  too, 
that,  notwithstanding  all  assertions  to  the  contrary,  the  great 
economy  of  fuel  of  the  modern  high  duty  pumping  engine  is  only 
realized  in  practice  when  it  is  working  steadily  at  its  maximum 
capacity.  A  varying  supply  is  fatal  to  its  best  results. 

In  a  word,  the  high  duty  pumping  engine  finds  its  most  fit- 
ting field  in  the  supplies  of  large  cities,  where  great  and  nearly 
constant  quantities  of  water  are  consumed  daily,  and  where  the 
facilities  for  repairs  and  the  procuring  of  skilled  labor  are  most 
abundant.  The  scale  descends  down  to  the  small  town  or  village, 
where  the  crudest  and  simplest  type  of  engine  suits  best  the 
case. 

The  actual  duty  of  a  pumping  engine  is  ascertained  by  a 
test.  This  test  consists  in  running  the  engine  and  pumps  for  a 
certain  number  of  hours,  and  during  that  period  measuring, 
weighing  and  timing  everything  that  is  done  by,  in  and  about 
the  entire  plant,  and  then  interpreting  the  data  thus  collected. 

The  duty  of  a  pumping  engine  is  best  expressed  by  the  num- 
ber of  foot  pounds  of  work  actually  performed  per  million  heat 
units  (B.  T.  U.)  delivered  by  the  boiler  to  the  engine.  Some- 
times, though  in  this  country  less  frequently  now  than  formerly, 
duty  is  expressed  in  foot  pounds  of  work  per  hundred  pounds  of 
coal  consumed.  The  objection  to  this  method  is  that  it  tests  coal, 
boiler  and  engine  as  a  whole,  whereas  their  performances  should 
properly  be  kept  distinct. 

The  principal  difficulty  in  these  tests  is  in  measuring  the 
volume  of  water  lifted.  When  the  quantities  are  comparatively 


WATER  SCPPLY   ENGINEERING.  131 

small  they  may  sometimes  be  measured  directly.  Weir  measure- 
ments come  next  in  order  of  merit.  Frequently  the  only  avail- 
able method  is  by  plunger  displacement,  allowance  being  made 
for  leakage  and  slip.  As  many  checks  as  possible  should  be  used 
in  determining  this  important  factor. 

The  volume,  or  weight  of  water  lifted  being  ascertained,  the 
next  thing  is  the  height  to  which  it  has  been  raised,  or  rather  the 
pressure  used  to  force  it  to  this  height.  Clearly  much  more  power 
would  be  required  to  force  one  million  gallons  100  ft.  high  in  a 
given  time,  through  a  pipe  8  in.  in  diameter  than  through  one  of 
16  in.  It  is  not,  therefore,  the  height  that  is  required  but  the 
actual  pressure  overcome  by  the  engine.  To  ascertain  this  two 
gauges  are  used,  a  pressure  gauge  on  the  force  main  and  a  vacuum 
gauge  on  the  suction,  unless  this  latter  is  under  a  head,  when  a 
pressure  gauge  is  applied  here  also.  These  pressures,  multiplied 
by  the  total  travel  of  the  piston,  give  the  foot  pounds  developed 
during  the  trial. 

The  total  heat  units  supplied  by  the  boiler  are  calculated  by 
taking  the  weight  of  all  the  water  fed  to  the  boiler  from  all 
sources,  and  the  difference  in  degrees  Fahrenheit  between  the 
initial  temperature  of  each  several  weight  of  water  and  the  total 
heat  of  the  dry  steam  delivered  at  boiler  pressure  to  the  cylinders. 

The  formula  for  duty  is  then: 

w  x  1000000 

Duty  = 

H 

In  this  equation  W=  total  work  in  foot  pounds,  and  H  •= 
total  heat  in  British  thermal  units: 

When  the  duty  is  estimated  in  work  done  in  foot  pounds 
per  100  Ibs.  of  coal  consumed,  the  formula  is : 

100  w 

Duty  = . 

c 

Where  C=  pounds  of  coal  consumed  during  the  time  that 


132  WATER   SUPPLY  ENGINEERING. 

the  work,   W,  is  being  performed.     To  convert  the  above  into- 
pounds  of  coal,  P,  per  hour  per  horse  power  : 


198 

T>     _.  _  __ 


Duty  in  mil.  ft.-pds. 


Thus  for  an  engine  having  a  duty  of  100  million  foot  pounds 
per  100  Ibs.  coal : 


198 

P  =  —  =  1.98, 
100 

or  practically  2  Ibs.  coal  per  hour  per  horse  power. 

Rating  1  Ib.  average  coal  as  equal  to  the  evaporation  of  10  Ibs. 
water  or  the  development  of  10,000  B.  T.  IL,  we  may  say  that 
"  high  duty"  implies  not  less  than: 

1,000,000  ft.  pds.  per  Ib.  coal 
100,000  "        water. 

100  "       B.  T.  U. 

Duty  trials  are  perhaps  most  generally  made  in  this  country 
under  actual  working  conditions,  that  is,  the  main  feed  is  pumped 
from  the  hot  well,  and  the  jacket  and  separator  water  fed  back  to 
the  boiler.  This  is  more  satisfactory  as  representing  normal  con- 
ditions, but  complicates  the  measurements. 

The  management  of  a  duty  trial  is  a  very  intricate  affair, 
and  cannot  be  fully  described  here.  The  report  of  the  commit- 
tee on  standard  method  of  conducting  duty  trials  of  pumping  en- 
gines, of  the  American  Society  of  Mechanical  Engineers,  in  its 
revised  form,  should  be  consulted  in  this  connection. 

As  illustrating  the  general  outline  :of  a  duty  trial,  the  follow- 
ing example,  condensed  and  simplified  from  one  given  in  the 
above  report,  will  now  be  instanced:  A  high  duty  compound 
pumping  engine  is  supplied  with  steam  at  135  Ibs.  absolute  pres- 
sure. This  corresponds  to  a  total  heat,  above  zero  Fahrenheit,  of 
1,220.70  B.  T.  U.  There  is  a  separator  on  the  main  steam  pipe. 
After  passing  through  this  separator  the  steam  is  found  to  still 
contain  H  per  cent. -moisture.  This  moisture  affects  the  latent 


WATER  SUPPLY   ENGINEERING.  133 

heat  of  the  steam  (which  at  above  pressure  is  866.60  B.  T.  U.), 
so  that  its  total  heat  above  zero  is  : 

1220.70  —  866.60  X  0.015  =  1207.7, 

Both  cylinders  are  jacketed,  and  there  is  a  reheater  supplied 
with  boiler  steam.  Water  from  jackets,  separator,  and  reheater 
feed  back  to  boiler.  The  different  supplies  of  water  fed  to  boiler 
during  trial  (10  hours),  with  their  temperatures,  are  as  follows  : 

Main  feed,  at  215° 18,863  Ibs. 

Low  pressure  jacket,  at  225° 615   " 

High  and  reheater,  at  290° 815   " 

Separator,  at  340° ... 210   " 

Total  feed 20,503  Ibs. 

The  total  heat  furnished  by  the  boiler  is  therefore  : 

Main  feed  (1,207.7  —  215)  18,863  = 18,725,300 

Low  pressure  jacket  (1,207. 7  —  255)  615  = . ... 604,361 

High  pressure  jacket  and  reheater  U.207.7  —  290)  815  = 747,926 

Separator,  neglected. 


Total  B.  T.  U 20,077.587 

The  net  area  of  pump  plunger  is  308  sq.  in.,  and  the  average 

stroke  3  ft. 

Number  of  single  strokes  during  trial 24,000 

Pressure  by  gauge  on  force  main 95.00  Ibs. 

vacuum  gauge  on  suction  main 3.69  " 

equivalent  to  difference  of  level  of  gauges..  4.31  " 

Total  pressure .103.00  Ibs. 

The  work  done  by  the  pump  is  therefore  308  x  103  x  3  x 
24000  =  2,284,1*8,000  ft.  Ibs. 

2284128000 

Duty  = =  113765066. 

20077587 

Indicated  horse  power;  as  determined  during  trial,  128.15. 
Pump  horse  power,  as  above  : 


Hence  : 


2284128000 

-  =  115.36. 

10  X  60  X  33000 


11536 

Efficiency  =  =  90  per  cent. 

128 .15 


134  WATER  SUPPLY  ENGINEERING. 

This  is  efficiency  of  the  engine  as  regards  work  done  by  the 
pistons,  not  as  regards  heat  utilized.  Upon  this  latter  basis,  the 
calculation  would  be  as  follows :  Since  1  B.  T.  U.  =  772  ft.  Ibs. 
and  one  horse  power  =  1,980,000  ft.  Ibs.  per  hour,  the 

1980000 

number   of   B.   T.   U.  per   hour    per  horse  power   is 

772 

=  2565.  But  the  total  B.  T.  U.  furnished  by  the  boiler,  per 
hour,  is  2007758.7.  Hence,  the  theoretical  horse  power  corre- 
sponding to  the  heat  unit  supplied  is  : 

<     20077587 

Theoretical  H.  P.  = =  782 

2565 

Therefore  : 

11536 
Efficiency  =   =  0.1475 

782 

The  above  theoretical  horse  power  is,  of  course,  impossible  of 
realization,  for  it  supposes  the  temperature  of  the  hot  well  to  be 
reduced  to  zero.  Suppose,  in  the  above  case,  for  round  numbers, 
that  the  temperature  of  the  hot  well  were  100°  ;  the  tempera- 
ture of  the  steam,  at  135  Ibs.  absolute  pressure,  is  350°  above  zero 
Fah.  and  810°  above  zero  absolute.  Then,  by  Carnot's  law,  the 
efficiency  of  a  perfect  heat  engine  working  between  the  given 
limits  is: 

350  -  100 


810 

Then,  according  to  one  view  of  the  subject  : 

1475 

Efficiency  = =  0.48  nearly. 

3086 

According  to  another  view,  which  considers  only  the  latent 
heat  of  the  steam  as  that  theoretically  utilizable,  we  have  for  the 
total  theoretically  possible  work-in  foot  pounds  per  pound  of 
steam  (or  feed  water),  the  following  formula  : 


w  = 

T  -f  46t) 


WATER  SUPPLY  ENGINEERING.  185 

In  which  : 

W  =  total  passible  work  in  foot  pounds  per  pound  of  feed  water. 
L  H  -  latent  heat  of  steam  at  given  pressure. 
T  =  temperature  of  steam  at  given  pressure. 
t  =  temperature  hot  well  or  condenser. 
T  -f  461  =  absolute  temperature  of  steam. 
772  =  Joule's  equivalent. 

In  the  example  : 

W  =  866.6   X  ?—  X  772  =  206510. 
810 

As  there  were  20,503  Ibs.  of  steam  furnished,  or  water 
fed: 

206510  X  20503  =  4234074530  ft.  Ibs. 

Comparing  this  with  the  work  actually  done  by  the  pumps, 
gives  : 

2284128000 

Efficiency  = =  0.54  nearly. 

4234074530 

The  usual  English  method  of  conducting  a  duty  trial  seems 
to  be  to  feed  the  boiler  exclusively  from  a  measured  tank,  wasting 
the  jacket  and  injection  water.  The  quantity  and  temperature 
of  the  wasted  discharge  of  the  air-pump  is  also  measured,  and 
the  quantity  of  heat  necessary  to  raise  it  from  its  initial  tempera- 
ture to  that  of  the  hot  well  calculated  and  recorded  as  "rejected 
heat."  This  quantity  of  heat,  reduced  to  B.  T.  U.  per  minute 
per  indicated  horse  power,  is  known  as  "  Donkin's  Coefficient/' 
When,  however,  the  feed,  injection,  and  jacket  water  are  measured 
separately,  a  more  accurate  estimate  is  possible  than  when  the  total 
discharge  of  air-pump  is  used.  To  this  rejected  heat  is  added 
the  heat  utilised  per  minute  per  indicated  horse  power,  which  is 

2565 

— —  =  42.75  multiplied  by  the  indicated  horse  power  developed 
60 

during  the  trial. 

The  rejected  and  utilized  heat  added  together  should  equal 
the  total  heat  reckoned  from  boiler  consumption.  It  always  falls 
short,  and  the  balance  is  put  down  to  errors  and  radiation. 


136  WATER  SUPPLY  ENGINEERING.. 

As  an  example  of  the  English  method  the  following  is  given 
based  upon  a  trial  made  by  Professor  Unwin.  The  wasted  jacket 
water  was  measured  separately,  and  a  more  accurate  basis  of  cal- 
culation thus  established  than  by  the  use  of  Donkin's  coefficient. 
All  calculations  are  made  in  this  example  for  the  total  duration 
of  trial  and  not  reduced  to  "  per  minute." 

Duration  of  trial  . .  24  hours. 

Total  water  pumped 200,000,000  Ibs. 

Lift,  including  friction  50  ft. 

Work  done 10,000,000,000  ft.  pds. 

Feed  at  51° 108,500  Ibs. 

Jacket  water  wasted 16,9001bs. 

Feed  used  in  work  (108,500-16,900) 91,600  " 

Injection  water,  at50° 3,633,000  " 

Condensed  steam 91,600  " 

Coalburned 11,000" 

Temperature  of  hot  well 75° 

"  steam,  75  Ibs.  absolute 307° 

Total  heat  of  above  steam     1208° 

Indicated  horse  power , 255.50 

Pump  horse-power 210.44 

210.44 
Efficiency =  82.3* 

255.50 

Rejected  heat  : 

Injection,  3,633,000  (75-50) 90,825,000 

Feed,  91,600  (75— 51  > 2,198,400 

Jacket  water,  16,900  (307-51) 4,326,400 

Total  rejected  heat  97,349,800 

Heat  units  utilized  during  trial,  255.5  X  2,565  X  24 15,728,580 

Total  heat  accounted  for...  113,078,380 

Total  heat  furnished  by  boilers,  108,500  (1,208-51)  =  125,534,500 

Heat  accounted  for  =  113,078  380 

Deficit  (about  102) 12,456,120 

The  duty,  calculated  per  million  heat  units  furnished  to  cyl- 
inders (108500  —  16900)  (1208  —  51)  is  : 

10000000000  X  1000000 

Duty  = • =  94356358  ft.  Ibs, 

91600  X  1157 

The  English  practice  is  to  calculate  duty  per  cwt.,  or  112  Ibs. 
of  coal.  Then  : 

10000000000  X  112 

Duty  = =  101818102  ft.  Ibs. 

11000 

Apart  from  minor  details,  and  considering  only  those  of  large 


WATER  SUPPLY  ENGINEERING.  137 

•capacity,  there  are  two  prominent  types  of  pumping  engines  in 
use  in  the  United  States ;  namely,  the  direct  acting,  with  or 
without  a  high  duty  attachment,  and  the  rotative,  or  crank  and 
flywheel  engine.  The  merits  of  these  two  types  are  fully  set 
forth  in  the  descriptive  pamphlets  of  their  respective  makers; 
here  it  will  suffice  to  briefly  enumerate  their  respective  claims,  as 
follows : 

The  advocates  of  the  direct-acting  type  claim  a  large  reduc- 
tion of  weight  by  replacing  (when  high  duty  is  required)  the  fly- 
wheel and  crank  shaft  by  a  comparatively  light  special  attach- 
ment, and  a  greater  security  against  damage  in  case  of  an  acci- 
dent suddenly  relieving  the  engine  of  its  loud  from  the  fact  that 
there  is  no  dangerous  momentum  stored  in  heavy  moving  parts. 
Also,  that  high  duty  is  realized  through  a  greater  range  of 
developed  power,  that  is,  nearly  the  same  economy  is  claimed 
when  working  at  reduced,  as  at  full  speed. 

The  advocates  of  the  rotative  type  claim  that  "  its  princi- 
pal advantages  are  positive  action  of  steam  valves  and  cut-offs, 
and  absolute  full  stroke  of  steam  pistons  and  plungers  under 
varying  pressures  of  steam  and  water.  In  these  engines,  there- 
fore, there  can  be  no  increase  in  the  clearance  spaces  between  the 
steam  pistons  and  cylinder  heads  causing  waste  of  steam,  nor  loss 
of  capacity  by  deficient  plunger  displacement." 

It  is  but  simple  justice  to  say  that  many  magnificent  speci- 
mens of  both  types  are  to  be  found  doing  excellent  service  all 
•over  the  country. 

ARCHES  AND  ABUTMENTS. 

Some  of  the  grandest  and  most  interesting  examples  of 
hydraulic  engineering  are  to  be  found  in  the  arched  aqueducts  of 
ancient  and  modern  times.  A  study  of  the  principles  of  the  arch 
is  therefore  an  essential  part  of  the  equipment  of  the  water- 
works engineer. 

The  span  being  given,  the  starting  point  of  all  arch  calcula- 


138 


WATER   SUPPLY    E>GINEERING. 


tion  is  thickness  or  depth  of  the  crown  at  the  key.  This  dimen- 
sion is  fixed  in  practice  by  some  one  of  the  various  empirical 
formulae  in  general  use.  Although  these  have  been  deduced  from 
existing  structures,  and  the  most  approved  ones  can  be  supported 
by  many  examples,  they  exhibit  considerable  variation  in  their 
results.  Here  are  five,  with  the  names  of  the  authorities  who- 
give  them: 

Perronnet.    D  =  I  +  0.035  S  (1) 

Croizette-Desnoyers.    D  =  0.50  +  0  38  \'R  (2) 

D  =  0.50  4  0.33   VR  (3) 


Boix.    D  =  0.75  VS 
Rebolledo.    D  =  1.15  +  0.035  (S  —  V} 


(4) 


In  these,  D  =  depth  of  key;  S  =  span;  R  =  radius  of  cnrva 
of  in  trades;  V=  versine  or  rise.  If  the  intrados  be  elliptical,  R  — 
assumed  or  approximate  radius  at  crown.  All  dimensions  in  feet. 

Of  these  formulae  (1)  is  said  by  Leville  to  apply  to  all  curves- 
of  intrados,  semi-circular,  segmental,  or  elliptical.  Formula  (2) 
applies  to  semi-circular  arches,  and  also  to  segmental  ones,  or 
those  formed  by  an  arc  of  a  circle  less  than  a  semi-circle,  when 

8 
the  rise  is  more  than  — .     If  less,  (3)  applies.     This   formula   is 

6 
slightly  changed  from  the  original. 

It  will  be  instructive  to  apply  these  formulas  to  a  series  of 
arches  of  different  spans,  for  the  purpose  of  'comparison.  Com- 
mencing with  semi-circular  arches  of  30,  60  and  100  ft.  span,  the 
following  table  gives  the  value  of  D  from  formula?  (1);  (2);  (4) 
and  (5). 


SEMI-CIRCULAR  ARCHES. 


S 

D 

(1) 

(2) 

(0 

(5) 

30 

2.05 

1.97 

2.33 

1.68 

50 

2.75 

2.40 

2.76 

2.03 

100 

4.50 

3.18 

3.48 

2.90 

WATER  SUPPLY   ENGINEERING. 


139 


For  segmental  arches  of  60°,  of  which  the  radius  is  equal  to 
the  span,  and  the  versine  or  rise  is  0.134  of  span  or  radius, 
formulae  (1),  (3),  (4)  and  (5)  give: 

SKGMENTAL  ARCHES  OF  60°. 


8 

D 

(1) 

(3) 

<4) 

(5) 

30 

2.05 

2.31 

2.33 

2.06 

50 

2.75 

2.83 

2.76 

2.67 

100 

4.50 

3.80 

3.48 

4.18 

These  results  agree  better  than  the  previous  ones,  but  it  is 
evident  that  we  can  exercise  considerable  latitude  in  the  matter 
of  keys  and  yet  have  good  authority,  and  still  better,  good  ex- 
amples to  sustain  us. 

These  thicknesses  of  key  apply  more  particularly  to  railroad 
and  highway  bridges.  They  are  sufficient  to  carry  on  the  arch 
ring  alone  and  without  taking  account  of  the  spandril  backing, 
track  and  trains,  with  2  or  3  ft.  of  earth  filling  over  the  extrados. 
For  heavier  embankments  some  authors  recommend  an  addition 
to  the  depth  of  key  of  2  per  cent,  of  height  of  embankment. 
Thus,  if  an  arch  carries  a  50-f t.  embankment,  add  one  foot  to 
depth  of  key. 

In  the  absence  of  special  empirical  formulae  for  aqueduct 
arches,  the  above  may  be  safely  taken  for  this  purpose,  especially 
since  for  aqueducts  the  spandril  backing  will  generally  be  carried 
up  level  with  the  extrados  at  the  crown. 

It  is  a  well-known  fact  that  a  semi-circular  arch  can  be  car- 
ried up  to  a  considerable  height  above  the  spring  line  before  the 
voussoirs  begin  to  bear  upon  the  centering,  as  they  are  kept  in 
place  below  this  height,  by  friction.  It  has  been  found  that  the 
point  at  which  centering  becomes  necessary  to  support  the  vous- 
soirs of  a  semi-circular  arch  occurs  at  about  half  the  height  of  the 
rise,  or  at  60°  measured  from  the  vertical.  The  joint  at  this 


140 


WATER  SUPPLY  ENGINEERING. 


point,  or  the  point  nearest  to  it,  is  known  as  the  "  joint  of  rup- 
ture," and  is  one  of  the  most  important  elements  of  arch  design- 
ing. It  is  up  to  this  level  that  the  backing  of  the  haunches  should 
always  be  carried.  The  arch  proper  commences  at  this  joint, 
and  includes  120°.  All  below  this  joint  must  be  considered  as 
forming  part  of  the  abutments. 

In  a  well-proportioned  voussoir  arch,  the  thickness  of  the 
arch  ring  should  increase  from  the  crown  toward  the  haunches. 
The  radial  length  of  any  joint  between  the  key  and  the  joint  of 
rupture  should  be  such  that  its  vertical  projection,  or  the  cosine 
of  the  angle  which  it  makes  with  the  vertical,  shall  be  equal  to 

D 

the  depth,  D,  of  the  key.     It  is,  therefore,  =  ,  a  being  the 

cos.  a 

angle  which  the  joint  makes  with  the  vertical.  The  cosine  of 
60°  (or  sine  of  30°)  being  ^2,  we  have  for  the  length,  L,  of  the 
joint  of  rupture  of  a  semi-circular  arch  : 

L  =  2D  (6) 

The  length  of  any  joint  intermediate  between  the  key  and 
joint  of  rupture  may  be  found  as  shown  on  the  right-hand  side  of 
Fig.  24,  by  drawing  the  horizontal  line  df&t  the  distance  c  d  = 


FIG.  24. 


D  from  the  springing  line  A  B.     Then  the  length  of  any  joint, 
a  b,  is  found  by  drawing  c  b,  and  taking  e  b    =  R. 


WATER  SUPPLY  ENGINEERING,  141 

This  process  can  be  continued  below  the  joint  of  rupture,  as 
shown  in  the  figure,  when  the  curve  of  the  extrados  rapidly  flat- 
tens, becoming  finally  an  asymptote  to  the  springing  line. 

The  above  process  is  due  to  Dejardin.  Dubosque  gives  a  more 
rapid  method,  resulting  in  a  somewhat  greater  thickness,  which  is 
shown  on  the  left-hand  side  of  Fig  24,  and  is  described  as  follows: 
Join  g  and  li,  the  exterior  extremities  of  the  joint  of  rupture  and 
the  imaginary  joint  at  the  crown.  Bisect  g  h  in  K.  Draw  Km 
perpendicular  to  g  h,  intersecting  the  vertical  at  m.  From  «?, 
with  radius  m  g,  m  h,  describe  the  arc  </  h  which  is  the  required  ex- 
trados. Dubosque  finishes  the  extrados  by  drawing  the  tangent 
g  n  to  intersect  with  the  back  of  the  abutment,  produced,  as 
shown.  This  process  applies  to  all  arches,  whether  semi-circular, 
segmental,  or  elliptical. 

Since  in  a  semi-circular  arch  the  joint  of  rupture  is  situated 
at  an  angle  of  60°  from  the  vertical,  it  follows  that  all  segmental 
arches  of  which  the  amplitude  is  equal  to  or  less  than  1^0°,  or 
in  other  words  in  which  quotient  of  the  span  divided  by  the  rise 
is  equal  to  or  greater  than  3.46,  have  their  joint  of  rupture  at 

D 

the  springing  line.      Since  the  length  of   any   joint  is , 

cosine  a 

the  length  of  skewback  joint  of  such  a  segmental  arch  can  be 
found  by  multiplying  the  depth  of  key  by  the  radius  of  intrados, 
and  dividing  it  by  the  radius  minus  the  rise,  thus  : 

DP 


R—  V 

Since  the  radius  of  a  circular  arc  is  given  by  the  relation  : 

8F~ 

we  have  : 


142  WATER  SUPPLY   ENGINEEBI^G. 

If  the  segmental  arch   has  an  amplitude  of  more  than  120°, 

8 
that  is  if  —  <  3.46  the    position   of  the    joint  of   rupture  is,  of 

course,  at  the  distance  of  half   the    radius  from    the    crown,  tho 
same  as  fora  semi-circular  arch. 

For  elliptical  or  false  elliptical  arches  the  joint  of  rupture 
occurs  at  half  the  rise,  the  same  as  for  semi-circular  ones,  but  its 
length  is  differently  determined.  Oroizette-Desnoyers  has  given 
the  following  series  of  coefficients  for  such  arches  for  different 

values  of  the  ratio  —  : 
V 

s 

-  =  3;  L  =  l.80D 

S  _ 
--4,  L 

S 

-  =  5  ;  L  =  1.40  D  (10) 

For  other  ratios  L  can  be  found  by  interpolation  : 
Arches  of  this  class  are  generally  of  the  "  false-elliptical " 
type,  that  is,  are  formed  by  an  odd  number  of  circular  arcs,  tan- 
gent to  each  other,  the  resultant  curve  closely  approximating 
a  true  ellipse,  three  being  the  smallest  number  of  such  arcs  that 
can  be  used.  This  is  called  a  "three-centred"  arch,  and  is  suit- 

8 
able  to  ratios  not   greater   than  —  =  3.     When  the   value  of  the 

V 
ratio  exceeds  3  a  greater  number  of  centres  should  be  used. 

The  two  radii,  in  terms  of  span  and  rise,  of  a  three- centred 
arch  are  given  by  Dubosque  : 

s  /          \ 

=  -  +  0.683(  S—  2rj  (11) 


WATER  SUPPLY   ENGINEERING.  143 

S  /  \ 

r  =  S—R  = 0.683  /S-2K)  (12) 

In  which  R  =  radius  of  large  central  arc,  and  r  =  that  of 
the  two  smaller  end  arcs. 

As  regards  top  thickness  or  thickness  at  the  spring  line,  T, 
of  abutments,  a  good  general  formula  is  given  by  Boix  : 

T=  1.30  +  0.11  S  \/—  03> 

V    V 

For  semi-circular  arches,  this  reduced  to  : 

T  =  1.30  +  0.20  S.  (U) 

Formulae  (13)  and  (14)  take  no  account  of  the  height  of  the 
abutment,  which  is  not,  contrary  to  what  might  be  supposed,  a  very 
important  factor.  If  the  top  thickness  be  determined  by  the  for- 
mula, the  batter  of  the  wall  will  amply  provide  for  any  additional 
thickness  rendered  necessary  by  the  greater  or  less  height. 

A  good  formula,  of,  I  believe,  German  or  Russian  origin,  for 
thickness  of  abutments  of  semi-circular  arches  in  which  the 
height,  H,  enters  is  : 

T  =  1  +  0.20  S  4-  0.16  H.  (15) 

In  ordinary  cases  this  may  be  reduced  to  : 

T  =  0.30  s. 
For  piers,  Rebolledo  gives  : 

T  =  2.50  D'+  0.10  H. 

In  which  D  =  depth  of  key  of  arch,  and  H  =  height  of 
pier.  In  wide  spans  and  heavy  loads,  the  weight  borne  by  foot 
of  piers  and  abutments  must  be  considered  with  regard  to  resist- 
ance to  crushing. 

EXAMPLE. — Determine  the   principal  dimensions  of   a  3-cen- 


144 


WATER  SUPPLY  ENGINEERING. 


tred  arch,  30-ft.  span,  10-ft.  rise,   abutments  12  ft.  high.     See- 
Fig.  25. 


From  (11)  or  (12). 


R  =  21.83  ft. 
r=8.17  ft. 


Thickness,  D,  of  key  from  formulae  (1)  to  (5)  respectively, 
2.05  ;  2.27  ;  2.33  ;  1.85.  Take  J9  =  2  ft.  Joint  of  rupture  oc- 

V 
curs  at  —  =  6  ft.  below  crown,  and  its  length,  per  (8)  =  1.80  D 

2 
=  3.60  ft.     The  top   thickness,    or   thickness   at  spring-line  of 

,30 

abutments  per  (13)  =  1.30  +  0.14  x  30i/—  =  8.57  ft. 

10 

The  formulae  for  thickness  of  abutments  give  that  necessary 
to  sustain  the  thrust  of  the  arch.  No  account  is  taken  of  the 
counter  thrust  of  the  earth  embankment  behind  the  abutment. 
In  many  cases  the  abutments  could  be  lightened  by  relieving 
arches  or  otherwise. 

When  an  arch  fails,  it  is  generally  on  account  of  settling  or 


WATER  SUPPLY   ENGINEERING.  145 

spreading  of  the  abutments,  or  to  bad  work  or  materials.  If 
there  is  no  movement  in  the  supports,  it  can  only  fail  by  direct 
crushing  of  the  materials — a  very  rare  case — or  by  distortion  of 
the  arch.  Distortion  is  caused  by  a  sinking  at  one  point  and 
rising  at  another.  The  object  is,  therefore,  to  get  such  a  sub- 
stantial and  evenly  distributed  permanent  load  upon  the  arch,  if 
possible,  before  striking  the  centres,  that  no  irregular  strains- 
can  come  upon  it  to  cause  distortion.  Naturally,  the  lighter  the 
permanent  road,  the  stronger  and  stiifer  must  be  the  arch  in 
order  to  resist  transient  and  unequal  loading.  These  remarks 
apply  more  particularly  to  viaducts,  for  an  aqueduct  arch  is  sel- 
dom subjected  to  unbalanced  stresses. 

As  regards  the  best  form  of  arch,  a  given  opening  can  be 
successfully  spanned  by  an  arch  of  any  form.  Generally  speak- 
ing, the  curve  of  intrados  is  selected  in  reference  to  the  special 
conditions  of  each  case,  the  amount  of  head  room  required,  etc^ 
There  are,  however,  certain  forms  which,  other  things  being: 
equal,  are  best  fitted  for  certain  kinds  of  loading.  If  an  arch  is 
to  sustain  a  single  load,  concentrated  at  the  centre  and  heavy  as- 
compared  with  the  weight  of  the  arch  itself,  the  gothic  arch  is 
most  suitable.  If  the  loading  increases  gradually  from  the  crown 
to  the  haunches,  as  in  the  case  of  an  ordinary  earthen  embank- 
ment, the  curve  of  pressure  would  more  nearly  approach  the  arc 
of  a  circle.  Should  the  loading  be  much  greater  at  the  haunches 
than  at  the  crown,  the  curve  would  approach  the  elliptical  form. 
In  all  cases  the  curve  will  be  found  to  rise  to  the  pressure  ;  for 
the  true  curve  of  pressure  is  always  represented,  inverted,  by 
a  flexible  cord,  similarly  loaded  to  the  arch.  Where  head  room  near 
haunches  can  be  spared,  and  great  economy  of  material  is  not 
essential,  the  full  centred,  or  semi-circular  arch,  will  generally 
be  preferred,  both  for  its  great  structural  stability  and  the  beauty 
of  its  proportions.  The  "High  Bridge"  of  the  old  Croton 
aqueduct  may  be  here  instanced.  When  the  full  centred  arch 
is  inadmissible,  the  segmental  arch,  or  that  formed  by  the  are 


146 


WATER  SUPPLY  ENGINEERING. 


of  a  circle  less  than  J80°  in  amplitude,  has  much  to  recommend 
it.  Of  these,  the  arc  of  60°.,  with  span  equal  to  radius  and 
rise  —  0.134  S  seems  a  very  happy  selection  both  as  to  appear- 
ance and  convenience  of  dimensions.  Should  an  arch  be  intended 
to  support  a  body  of  water  in  direct  contact  with  its  extrados,  the 
proper  theoretical  form  would  be  that  of  the  hydrostatic  arch, 
which  resembles  a  cycloid,  which  in  turn  resembles  an  ellipse. 
In  all  practical  cases,  however,  such  as  tanks  or  aqueducts,  a 
level  bottom  is  necessary,  and  the  spandrils  are  built  up  level 
with  the  crown.  In  this  case  the  conditions  governing  the  hydro- 
static curve  do  not  obtain,  and  the  load  from  the  water  is  simply 
an  equally  distributed  one,  pressing  at  all  points  vertically  down- 
ward, or  at  least  practically  so. 

Although  the  above  empirical  formulae  suffice  to  correctly 
proportion  any  except  very  unusual  forms,  it  will  be  well  to  de- 
vote a  few  words  to  the  more  theoretical  features  of  the  subject. 

Suppose  it  were  wished  to  calculate  the  curve  of  pressure  in 
the  semi-circular  voussoir  arch,  half  of  which  is  shown  in  Fig.  26. 


FIG.  26. 


Assuming  it  to  be  of  good  materials  and  workmanship,  without 
which  all  calculation  would  be  impossible,  that  part  of  the  struc- 


WATER  SUPPLY  ENGINEERING.  147 

tn  re  lying  above  the  joint  of  rupture  e/is  taken  as  comprising 
the  arch  proper.  The  section  is  supposed  to  be  divided  into  an 
arbitrary  number  of  fictitious  voussoirs  a,  b,  c  and  d,  in  the 
figure,  which  will  give,  equally  well  with  the  true  ones,  the  form 
of  the  curve  of  pressure.  The  weight  of  and  upon  each  of  these 
voussoirs  is  then  estimated,  and  the  position  of  the  line  passing 
through  the  centre  of  gravity  of  each  voussoir  and  its  load  deter- 
mined. From  these  data  the  position  of  the  vertical  line  JF pass- 
ing through  the  centre  of  gravity  of  the  entire  section  above  joint 
of  rupture  is  fixed.  The  horizontal  line  If  cutting  the  central 

joint  at  one  third  f  ^- — J  of  its  length  from  the  top  is  then  drawn 

to  its  intersection  with  the  vertical  W.  From  the  point  of  in- 
tersection the  line  Z  is  drawn  cutting  the  joint  of  rupture  at 

one  third  t —  J  of  its  length  from  the  bottom.      The  weight 

of  the  half  arch  and  load  above  ef  is  then  laid  off  to  scale  on  W, 
and  from  its  extremity  the  horizontal  line  g  h  is  drawn  to  Z, 
completing  the  triangle  of  forces.  The  length  of  the  line  g  h  to 
the  same  scale  as  JF  gives  the  value  of  horizontal  thrust  at  the 
crown.  On  the  horizontal  line  H  produced,  take  aj=  g  h,  and 
from  it  extremity  lay  off/  K  =  weight  of  voussoir,  a,  and  load. 
Join  a  k.  On  a  Tc  produced,  lay  off  b  I  =  a  k.  From  I,  lay  off 
lm  =  weight  of  voussoir,  b,  and  load.  Proceed  thus,  working 
the  curve  down  from  voussoir  to  voussoir  to  the  line  of  rupture. 
The  last  resultant  should  coincide  with  the  oblique  line  Z,  which 
fact  furnishes  an  excellent  check  upon  the  accuracy  of  the  work. 
The  broken  line  representing  the  line  of  pressure  can  now  be 
harmonized  with  a  curve,  drawn  by  hand  and  the  actual  voussoirs 
laid  down  on  the  drawing  to  show  where  and  at  what  angle  the 
curve  cuts  their  joints. 

The  line  of  pressure  may  be  continued  down  to  the  spring- 
ing line  and  on  through  the  abutment,  but  this  is  not  generally 
necessary. 


148  WATER  SUPPLY  ENGINEERING. 

The  reason  why  the  points  of  application  of  If  and  Z  at  the, 
crown  and  joint  of  rupture  are  placed  at  the  upper  and  lower 
extremities,  respectively,  of  the  middle  third  of  these  joints  is 
the  following:  The  general  tendency  of  all  arches,  except  those 
carrying  very  unusual  loads  at  the  haunches,  is  to  sink  at  the 
crown  and  consequently  rise  at  the  haunches.  When  the  crown 
sinks  the  arch  opens  at  the  intrados,  in  the  neighborhood  of  the 
key,  rotating  upon  its  upper  edge,  at  the  extrados.  Inversely, 
the  joint  of  rupture  will  open  at  the  extrados,  rotating  around  its 
lower  edge,  at  the  intrados.  If  the  joint  at  the  crown  does  not 
actually  open,  there  is  always  a  tendency  to  do  so,  and  a  corre- 
sponding tendency  at  the  joint  of  rupture  ;  and  at  the  points 
around  which  the  voussoirs  tend  to  rotate  the  compressive  stresses 
are  at  their  maximum,  diminishing  progressively  until  reaching 
the  other  extremity  of  the  joint,  where  the  tendency  is  to  open. 
At  this  point  the  compressive  stresses  become -zero.  The  total 
compressive  stresses  on  these  joints  may  therefore  be  represented 
graphically  by  the  area  of  a  right-angle  triangle  constructed  upon 
each  joint,  with  its  base  at  the  extremity  around  which  rotation 
tends  to  take  place,  and  its  apex  at  the  extremity  which  tends  to 
open.  The  resultant  stress  passes  through  the  centre  of  gravity 
of  each  triangle,  which  is  at  one-third  of  its  height  from  the 
base.  The  heights  of  the  triangles  being  represented  by  the 
length  of  the  joints,  the  points  of  application  of  H  and  Zmust, 
to  conform  with  the  above  theory,  be  placed  as  in  the  figure.  In 
this  connection  see  pages  87  and  88. 

For  a  properly  proportioned  arch,  such  as  would  be  the  out- 
come of  the  practical  rules  already  laid  down,  and  is  shown  in 
Fig.  26,  the  amount  of  horizontal  thrust  at  the  crown — which  is 
to  arch  calculation  what  abutment  reaction  is  to  that  of  girders 
and  bridges — can  be  obtained  more  readily  than  in  the  above  ex- 
ample by  the  use  of  Navier's  formula  : 

H  =  P  X  R  (17) 


WATER  SUPPLY   ENGINEERING.  149 

in  which  P  =  pressure  or  weight  per  square  unit  at  the  crown, 
and  R  =  radius  of  the  intrados  at  the  crown,  expressed  in  the  same 
linear  unit.  Thus,  in  last  example,  Fig,  26,  if  the  weight  of  arch 
and  load  on  or  in  the  immediate  vicinity  of  the  key  were  1,000 
Ibs.  per  square  foot,  and  the  radius  15  ft.,  the  total  horizontal 
thrust  would  =  15,000  Ibs.  In  example,  Fig.  25,  if  the  unit 
pressure  were  the  same,  the  radius  at  the  crown  being  21.84  ft  , 
the  total  horizontal  thrust  H  would  equal  21, 840  Ibs. 

As  regards  the  whole  subject  of  arch  design,  it  may  be  broadly 
stated,  on  the  authority  of  Dejardin,  that  if  the  arch  is  propor- 
tioned according  to  the  rules  already  laid  down,  and  is  well  con- 
structed with  good  materials,  no  calculation  whatever  is  needed 
to  demonstrate  its  stability.  If,  however,  the  engineer  should  be 
called  upon  to  discuss  an  existing  structure  built  upon  different 
lines  (and  which,  nevertheless,  might  fulfil  all  the  requirements 
of  stability)  he  should  proceed  as  above  directed,  making  differ- 
ent assumptions  until  a  line  of  pressure  is  found  which  shall 
at  least  lie  inside  of  the  arch,  at  all  points,  the  presumption  being 
that  if  such  a  line  can  exist,  the  arch  will  find  it.  Should  no 
curve  be  found  which  did  not  cut  the  intrados  or  extrados  at  some 
point  or  points,  it  would  indicate  that  rotation  would  occur  around 
such  point  or  points,  with  intense  local  compressive  stress.  If 
this  result  were  found  in  any  design  under  discussion,  it  would 
justify  the  rejection  of  such  design.  If  it  were  found  in  any  ex- 
isting structure,  it  would  prove  that  the  construction  and  ma- 
terials of  the  arch  were  of  a  nature  to  permit  it  to  resist  bending 
moments  at  such  points,  or  that  the  assumed  data  regarding 
weights  and  loading  were  incorrect.  Should  the  line  of  pressure, 
while  not  leaving  the  limits  of  the  section,  approach  very  near  to 
the  intrados  or  extrados,  the  amount  of  compressive  stress  set  up 
at  these  points  is  to  be  determined  according  to  the  rules  laid 
down  for  masonry  dams,  considering  the  compression  to  be  con- 
centrated upon  the  area  of  joint  lying  between  the  point  of  ap- 
plication of  the  pressure  and  the  nearer  extremity  of  the  joint. 


150 


WATER  SUPPLY  ENGINEERING. 


The  chief  uncertainty  which  militates  against  the  value  of  all 
arch  calculation  lies  in  the  fact  that  we  are  obliged  in  most  cases 
to  make  almost  random  guesses  at  the  superincumbent  loads  ac- 
tually and  not  theoretically  sustained  by  the  arch.  As  a  simple 
example  of  this  may  be  cited  the  case  of  an  arch  sustaining  a 
high  masonry  wall.  Theoretically,  the  whole  weight  of  the  wall 
would  be  resting  upon  the  arch,  and  the  higher  the  wall  the  more 
danger  to  the  arch.  Practically,  we  know  that  the  higher  the 
wall  the  less  danger  would  there  be  of  its  falling  in  if  we  should 
break  an  opening  through  it  near  the  bottom. 

It  will  be  noticed  in  all  that  precedes  that  the  joint  of  rup- 
ture has  been  placed  at  half  the  height  of  the  rise.  It  is  found 
in  practice  that  this  assumed  position  results  in  a  well-propor- 
tioned and  secure  arch,  particularly  if  the  masonry  backing  be 
carried  up  to  this  height  at  the  extrados.  Strictly  speaking, 
however,  we  cannot  so  broadly  generalize  the  problem,  and  prob- 
ably it  is  impossible  to  tell  exactly  where  the  joint  of  rupture  is 
located  in  any  existing  arch.  Lame  and  Olapeyron  assert  that 


FIG.  27. 


the  line  of  rupture  a  b,  Fig.  27,is  so  situated  that  the  tangent  c  d, 
at  the  point  b,  and  the  tangent  e/at  the  crown,  will  always  inter- 


WATER  SUPPLY  ENGINEERING.  151 

sect  at  the  line  g  It,  passing  through  the  centre  of  gravity  of  the 
mass  lying  above  the  line  of  rupture  in  such  a  position  that  these 
three  lines  will  all  meet  at  one  and  the  same  point,  i.  To  deter- 
mine the  position  of  the  joint  of  rupture  by  this  rule,  it  would 
be  necessary  to  operate  by  trial  and  error.  It  might  be  used  to 
test  the  assumed  position  of  the  joint  in  any  particular  case,  but 
it  does  not  seem  to  possess  sufficient  practical  utility  to  render  its 
use  recommendable. 

All  that  precedes  relates  to  the  voussoir  arch,  the  funda- 
mental principle  of  which  is  that  it  preserves  its  stability  by 
equilibrium  alone,  independent  of  any  cohesion  of  the  mortar  in 
which  the  stones  are  bedded.  Brick  and  concrete  arches  belong 
to  a  totally  different  class,  as  their  stability  depends  principally  or 
wholly  upon  the  cohesion  of  the  mortar.  They  are  monolithic  in 
character,  and  while  the  empirical  formulae  established  for  voussoir 
arches  can  be  used  to  determine  their  proper  dimensions,  the  cal- 
culations respecting  the  line  of  pressure  do  not  apply. 

The  increased  thickness  toward  the  haunches  of  a  brick  arch 
may  be  given  by  bonding  in  additional  rings,  or,  as  will  generally 
be  found  preferable,  by  giving  the  arch  a  uniform  thickness 
throughout,  and  obtaining  the  desired  increase  by  adding  to  the 
spandril  backing,  as  shown  in  Fig.  28.  In  the  case  of  brick  arches 
of  small  span,  ib  is  best  to  build  them  in  independent  concentric 
rings,  but  if  the  span  and  radius  of  curvature  are  relatively  large, 
it  will  frequently  be  found  advisable  to  bond  the  rings  together. 

Should  it  be  desired  to  apply  calculation  to  a  brick  arch  in 
order  to  ascertain  its  probable  stability,  recourse  should  be  had 
to  the  old  method  of  Lame  and  Glapeyron,  which  is  based  upon 
Boistard's  experimental  researches.  Without  going  into  a  full 
description  of  this  method,  which  can  be  found  in  text  books, 
the  formula  will  be  given  to  determine  whether  a  given  arch  is 
stable  as  against  sinking  at  the  crown  and  raising  at  the 
haunches,  by  far  the  most  common  case  of  failure. 


152 


WATER  SUPPLY  ENGINEERING. 


Referring  to  Fig.  28,  supposing  the  abutments  to  be  unyield- 
ing, and  considering  only  the  arch  a  b  c  d  e  f,  exclusive  of  the 


backing,  the  notation  is  as  follows,  a  b  being  the  springing   line 
and  c  d  the  assumed  joint  of  rupture: 

W  =  weight  of  mass  ab  c  def  including  loading. 

W  =  weight  of  mass  c  d  ef  including  loading. 

A     =  distance  from  a  to  line  passing  through  centre  of  gravity  of  W. 

x      =  distance  from  d  to  line  passing  through  centre  of  gravity  W. 

y     —  vertical  distance  e  g,  from  d  to  e. 

JR    =  total  rise,  o  e,  from  springing  line  to  extrados. 

Then  the  equation  for  exact  static  equilibrium  is  : 
and  for  stability  : 


(18) 


W  A  —W  R  —  >  o 


(19) 


Thus,  in  the  arch  shown  in  Fig.  28,  assuming  a  span  of  30 
ft.;  a  rise  of  10  ft.,  and  a  depth  of  key  =  2.50  ft.,  let  the 
data  be  :  >  • 


W  -  6,000  Ibs. 
W  =  2,250  Ibs. 

R  =  12.50  ft. 
A  =  7.40  ft. 

x  =  5.40  ft. 

y  =  7.50  ft. 


WATER  SUPPLY  ENGINEERING.  153 

Then  from  (18)  and  (19): 

5.4 

6000  X  7.40  -  2250  X  12.50  X =  24,150  Ibs. 

7.5 

This  would  indicate  that  the  moment  W  A,  making  for 
stability,  had  a  factor  of  safety  of  nearly  1.85. 

It  will  be  noted  that  equations  (18)  and  (19)  contain  two 
variables,  depending  upon  the  position  of  the  joint  of  rup- 
ture. Write  (18)  in  this  form: 

iW A-  W  x\ 
R  ( -  J  =  o  (20) 

Then,  should  the  arch  break,  it  would  be  at  some  joint,  a  #, 

W  x 
such  as  would  make          ~  maximum.      This  point  can  be  found 

y 

by  trial  and  error.  If  such  maximum  value  gives  a  negative 
result  in  (20),  it  would  indicate  that  the  portion  of  the  arch 
below  the  joint  of  rupture  was  too  light. 

It  is  necessary  that  the  constructing  engineer  should  be 
familiar  with  the  above  processes  of  calculation,  or,  at  least,  be 
aware  of  their  existence,  in  order  to  know  what  can  and  cannot 
be  accomplished  by  figuring.  The  successful  development  of 
arch  building  having  been  mainly  along  purely  experimental 
lines,  very  little  aid  is  to  be  hoped  for  from  abstract  mathematical 
reasoning,  for  the  very  good  reasons — among  others — that  the 
most  important  data  must  be  assumed,  or.  in  other  words, 
guessed  at,  and  that  in  the  case  of  voussoir  arches  the  adhesion 
and  cohesion  of  the  rnortar  is  and  must  be  ignored,  although  it 
may  really  play  a  very  important  part,  particularly  where  small 
materials  are  used.  In  the  case  of  concrete  arches,  when  the 
limit  of  small  and  amorphous  materials  is  reached,  the  strength 
of  the  in  or  tar  is  everything. 

The  first  requisites  for  an  arch  are  unyielding  foundations 
and  abutments,  without  which  fracture  and  perhaps  destruction 
must  ensue.  Then  come  good  workmanship  and  materials,  and 


154  WATER  SUPPLY   ENGINEERING. 

great  judgement  as  to  time  and  manner  of  striking  centres  and 
loading  arch.  Design  has  been  purposely  left  to  the  last,  because 
if  all  the  other  requirements  are  observed,  the  shape  and  dimen- 
sions of  the  mere  arch  ring  are  matters  which,  though  important, 
are  still  of  lesser  moment. 

For  further  details  of  arch  design,  see  "  Van  Nostrand's 
Engineering  Magazine/' for  December,  1883,  and  February,  1884. 

EXPLANATION  OF  THE  TABLES. 

Table  1  gives  the  areas  of  circles  in  square  feet,  correspond- 
ing to  diameters  in  inches. 

Table  II  contains  five  columns,  headed  respectively  Z>,  F, 
Q,  G  and  //  P,  giving  various  properties  of  rough  cast-iron 
long  pipes  of  different  diameters  having  falls  of  one  to  ten  per 
thousand.  D  =  diameter  of  pipe  in  inches;  V  —  velocity  in  feet 
per  second;  Q  —  discharge  in  cubic  feet  per  second;  G  =  dis- 
charge in  U.  S.  gallons  per  hour,  and  II P  =  theoretical  or  net 
horse-power  necessary  to  raise  the  quantity  discharged  one  foot 
high.  In  this  table  Fis  calculated  by  the  formula  (1  bis.)  or 
(1  ter.},  Q  is  calculated  by  multiplying  Fby  the  area  of  the  pipe, 
G  by  multiplying  Q  by  27,000,  and  H  P  either  by  dividing  Q  by 
8.82  or  multiplying  G  by  42  and  pointing  off  seven  decimal 
places. 

EXAMPLE. — A  pipe  40  inches  in  diameter  is  laid  to  a  grade  of 
i-oVo-  What  is  the  discharge  and  the  net  horse-power  necessary 
to  lift  the  volume  discharged  to  a  height  of  113  ft.?  The  dis- 
charge is  947,241  U.  S.  gallons  per  hour/and  the  horse-power 
for  each  foot  of  lift  is  3.98.  Then  3.98  x  113  =  449.74  is  the 
required,  net  horse-power. 

Any  of  the  above  data  can  be  obtained  for  falls  per  thousand 
not  given  in  the  table  by  multiplying  the  values  given  for  falls- 
of  .j-^-y.  by  the  square  root  of  the  fall. 

EXAMPLE. — A  pipe  20  in.  in  diameter  has  a  fall  of  3.43  per 


WATER  SUPPLY  ENGINEERING. 


155 


1,000.  What  is  the  discharge  in  cubic  feet  per  second  ?  What 
the  horse  power  per  foot  of  lift  ?  I1  or  a  fall  of  ^n>  Q  =  3.511. 
Therefore  with  a  fall  of  3,43,  Q  =  3 .5 11 4/3 .43  =  6.50  cu.  ft. 

per  second.  Again,  for  115Vo>  HP=  0.40.  Therefore,  for 
.ViS,  HP  =  0.40  ,y/^43  =  0.74. 

RI 

Table  III.  gives   the  value  of  — —  for  different  values  of  R, 

U* 

from  which  the  mean  velocity  U  can  be  deduced. 

EXAMPLE. — Fig.  29  represents  the  cross-section  of  the  old 


FIG.  29. 

Croton  Aqueduct.  When  running  to  within  1.12  ft.  of  the 
crown,  as  shown  in  the  figure,  the  wet  section  =  49.19  sq.  ft., 
the  wet  perimeter  20.72  ft.,  and  the  mean  hydraulic  radius 

49.19 
R    = =  2.37.     Let  /  =  0.00021.     Then  by  table  III. : 

20.72 


U' 


=  0.00006794. 


156  WATER  SUPPLY  ENGINEERING. 

Also  by  the  data  : 


72  / 


Therefore  : 


=  0.0004977. 
F2 


0.0004977 

=  0.00006794 


49770 
~  6794 
U  =  2.71. 


This  value  agrees  quite  nearly  with  that  obtained  by  experi- 
ments with  floats  executed  under  the  author's  direction  in  1884. 

It  will  be  readily  seen  how  extremely  useful  this  table  is  in 
rapidly  determining  the  discharge  of  conduits. 


TABLE  I. 


D  =  inches. 

A  =  square  feet. 

D  -  inches. 

A  =  square  feet. 

1 

0.00545 

26 

3.6868 

'2 

0.02180 

27 

3.9760 

3 

0.0491 

28 

4.2760 

4 

0.0872 

29 

4.5868 

5 

0.1364 

30 

4.9087 

6 

0.1964 

31 

5.2413 

7 

0.2672 

32 

5.5848 

8 

0.3490 

33 

5.9394 

9 

0.4418 

34 

6.3048 

10 

0.5454 

35 

6.6812 

11 

0.6599 

36 

7.0686 

12 

0.7854 

37 

7.4665 

13 

0.9217 

38 

7.8756 

14 

1.0690 

39 

8.2957 

15 

1.2272 

40 

8.7264 

16 

1.3962 

41 

9.1682 

17 

1.5762 

42 

9.6211 

18 

1.7671 

43 

10.0847 

19 

1.9689 

44 

10.5589 

20 

2,1816 

45 

11.0440 

21 

2.4052 

46 

11.5408 

22 

2.6397 

47 

12.0479 

23 

2  8852 

48 

12.5660 

24 

3.1416 

49 

13.0951 

25 

3.4087 

50 

13.6354 

WATER  SUPPLY  ENGINEERING. 


157 


TABLE    II. 


1000 


D 

V 

Q 

G 

HP 

3 

0.56 

0.027 

729 

0.003 

4 

0.66 

0.057 

1539 

0.007 

6 

0.83 

0.163 

4401 

0.02 

8 

0.99 

0.344 

9288 

0.04 

10 

1.12 

0.612 

16524 

0.07 

12 

1.23 

0.966 

26082 

0.11 

U 

1.34 

1.432 

38664 

0,16 

16 

1.44 

2.010 

54270 

0.23 

18 

1.53 

2.704 

73008 

0.31 

20 

1.61 

3.511 

94797 

0.40 

22 

1.70 

4.488 

121176 

0.51 

24 

1.77 

5.561 

150147 

0.63 

26 

1.84 

6.784 

183168 

0.77 

28 

1.91 

8.167 

220509 

0.93 

30 

1.99 

9.769 

263763 

1.11 

32 

2.06 

11.505 

310635 

1.31 

34 

2.12 

13.367 

360909 

1.52 

36 

2.20 

15.552 

419904 

1.76 

38 

2.26 

17.800 

480600 

2.02 

40 

2.32 

20.247 

540669 

2.30 

42 

2.38 

22.898 

618246 

2.60 

44 

2.43 

25.658 

692766 

2.91 

46 

2.49 

28.737 

775899 

3.26 

48 

2.54 

31.918 

861786 

3.62 

1000 


D 

V 

Q 

G 

HP 

3 

0.79 

0.039 

1053 

0.005 

4 

0.93 

0.081 

2187 

0.010 

6 

1.18 

0.231 

6237 

0.03 

8 

1.40 

0.489 

13203 

0.06 

10 

1.59 

0.868 

23436 

0.10 

12 

1.74 

1.366 

36882 

0.16 

14 

1.90 

2.031 

54837 

0.23 

16 

2.05 

2.862 

77274 

0.33 

18 

2.16 

3.817 

103059 

0.43 

20 

2.28 

4.973 

134271 

0.56 

22 

2.40 

6.336 

171072 

0.72 

24 

2.50 

7.855 

212085 

0.89 

26 

2.60 

9.586 

258822 

1.09 

28 

2.70 

11.545 

311715 

1.31 

30 

2.81 

13.794 

372438 

1.56 

32 

2.91 

16.252 

438804 

1.84 

34 

3.00 

18.915 

510705 

2.15 

36 

3.11 

21.985 

593595 

2.49 

38 

3.20 

25.203 

(580481 

2.86 

40 

3.28 

28.625 

772875 

3.25 

42 

3.37 

32.423 

875421 

3.68 

44 

3.44 

36.323 

980721 

4.12 

46 

3.52 

40.624 

1096848 

4.61 

48 

3.59 

45.112 

1218024 

5.12 

158 


WATER  SUPPLY  ENGINEERING. 


TABLE  If.  — (CONTINUED.) 


1000 


D 

V 

3 

a 

HP 

3 

0.97 

0.048 

1296 

0.006 

4 

1.15 

0.100 

2700 

0.012 

6 

1.44 

0.282 

7614 

0.03 

8 

1.72 

0.600 

16200 

0.07 

10 

195 

1.065 

28755 

0.12 

12 

2.13 

1.672 

45144 

0.19 

14 

232 

2.480 

66960 

0.28 

16 

2.51 

3.504 

94608 

0.40 

18 

2.65 

4.683 

126441 

0.53 

20 

279 

6.085 

16429o 

0.69 

22 

2.93 

7.735 

208845 

0.88 

24 

3.0o 

9H15 

259605 

1.09 

26 

3.19 

11761 

317547 

1.33 

28 

3.31 

14.154 

382158 

1.61 

30 

3.45 

16936 

457272 

1.92 

32 

3.57 

19.938 

538326 

2.26 

34 

367 

23.139 

624753 

2.62 

36 

3.81 

26.933 

727191 

305 

38 

3.91 

30.795 

831465 

3.49 

40 

4.02 

35  083 

947241 

3.98 

42 

4.12 

39639 

1070253 

4.50 

44 

4.21 

44.453 

1200231 

5.04 

46 

4,31 

49742 

1343034 

5.64 

48 

4.40 

55290 

1492830 

6.27 

1000 


D 

V 

Q 

a 

HP 

3 

1.12 

0.055 

148) 

0.006 

4 

1.32 

O.U5 

3iOi 

0.013 

6 

1.67 

0.^27 

8S29 

0.04 

8 

1.98 

u.691 

18657 

0.08 

10 

2.25 

1.2-*) 

33185 

0.14 

12 

2.46 

1.931 

5213f 

0.22 

14 

2.68 

2.005 

77355 

0.33 

16 

2.90 

4.048 

109298 

0.46 

18 

3.06 

5.407 

145989 

0.61 

20 

3.22 

7.023 

189621 

0.80 

22 

3.40 

8.976 

242352 

1.02 

24 

3.54 

11.123 

300321 

1.26 

26 

3.68 

13.568 

366336 

1.54 

28 

3.82 

16.334 

441018 

1.85 

30 

3.98 

19.538 

527526 

2.22 

32 

4.12 

23.010 

621270 

2.61 

34 

4.24 

26.733 

721791 

3.03 

36 

4.40 

31.104 

8:^9808 

3.53 

38 

4.52 

35.600 

961200 

4.04 

40 

4.6t 

40.493 

1093311 

4.59 

42 

4.76 

45.796 

1236492 

5.19 

44 

4.86 

51.317 

1385559 

5.82 

46 

4.98 

57.474 

1551798 

6.52 

48 

5.08 

63.835 

1723545 

7.24 

WATER  SUPPLY  ENGINEERING. 


159 


TABLE  II.— (CONTINUHD.l 


1000 


D 

V 

Q 

G 

HP 

3 

1.25 

0061 

1647 

0.007 

4 

148 

0129 

3483 

0.015 

6 

1.86 

0.365 

9855 

0.04 

8 

2.23 

0.778 

21006 

0.09 

10 

2.51 

1.370 

36990 

0.16 

12 

2.75 

2.159 

5*293 

0.25 

14 

3.00 

3207 

86589 

0.36 

16 

3.24 

4.523 

122121 

0.51 

18 

3.42 

6.043 

163161 

0.69 

20 

3.60 

7.852 

212004 

0.89 

22 

3.79 

10.006 

270162 

1.14 

24 

3.96 

12442 

335934 

1.41 

26 

4.11 

15.154 

409158 

1.72 

28 

4.27 

18.259 

492993 

2.07 

30 

4.45 

21.845 

589815 

2.48 

32 

4.61 

25747 

695169 

2.92 

34 

474 

29.886 

806922 

3.39 

36 

492 

34779 

939033 

3.94 

38 

505 

39.774 

1073898 

4.51 

40 

5.19 

45.293 

1222911 

5.14 

42 

5.32 

51.184 

1381968 

5.80 

44 

5.43 

57.335 

1548045 

6.50 

46 

5.57 

64  283 

1735641 

7.29 

48 

568 

71375 

1927125 

8.09 

1000 


D 

V 

Q 

G 

HP 

3 

1.37 

0.067 

1809 

0.008 

4 

1.62 

0141 

3807 

0.016 

6 

2.04 

0400 

10800 

0.05 

8 

2.44 

0.852 

23004 

0.10 

10 

2.75 

1.502 

40554 

017 

12 

3.01 

2.363 

63801 

0.27 

14 

3.28 

3.51)6 

94662 

0.40 

16 

3.55 

4.956 

133812 

0.56 

18 

3.75 

6.626 

178902 

0.75 

20 

3.94 

8.593 

v  32011 

O.bS 

22 

4.15 

10.956 

295812 

1.24 

24 

4.33 

13.605 

367335 

1.54 

26 

4.50 

16.592 

447984 

i.bs 

28 

4.68 

20.012 

540324 

2.27 

30 

4.87 

23.907 

645489 

2.71 

32 

5.04 

28.148 

759996 

3.19 

34 

5.19 

32.723 

8835J1 

3.71 

36 

5.39 

38.102 

1028754 

4.32 

38 

5.53 

43.554 

1175958 

4.94 

40 

5.68 

49.c69 

1338363 

5.62 

42 

5.83 

56.09-  » 

1514430 

6.36 

44 

5.95 

62.826 

1696302 

7.13 

46 

6.10 

70.400 

19008'  in 

7.98 

48 

6.22 

78.161 

2110347 

8.86 

160 


WATER  SUPPLY  ENGINEERING. 


TABLE  II.— (CONTINUED.) 


moo 


D 

V 

Q 

G 

HP 

3 

1.48 

0.073 

1971 

0.008 

4 

1.75 

0.152 

4104 

0.017 

6 

2.20 

0.431 

11637 

0.05 

8 

2.62 

0.914 

24678 

0.10 

10 

2.97 

1.619 

43713 

0.18 

12 

3.26 

2.559 

69093 

0.29 

14 

3.55 

3.795 

102465 

0.43 

16 

3.84 

5.361 

144747 

061 

18 

4.05 

7.156 

193212 

0.81 

20 

4.26 

9.295 

250965 

1.05 

22 

4.48 

11.827 

319329 

1.34 

24 

4.68 

14.705 

397035 

1.67 

26 

4.87 

17.956 

484812 

2.04 

28 

5.05 

21.594 

583038 

2.45 

30 

5.27 

25.870 

698490 

2.93 

32 

5.45 

30.438 

821826 

3.45 

34 

5.61 

35.371 

955017 

4.01 

36 

5.82 

41.142 

1110834 

4.67 

38 

5.98 

47.098 

1271646 

5.34 

40 

6.14 

53.578 

1446606 

6.08 

42 

6.30 

60.612 

1636524 

6.87 

44 

6.43 

67.894 

1833138 

7.70 

46 

6.58 

75933 

2050191 

8.61 

48 

6.72 

84.444 

2279988 

9.58 

1000 


D 

V 

Q 

G 

HP 

3 

1.58 

0.077 

2079 

0.009 

4 

1.87 

0.163 

4401 

0.019 

6 

2.36 

0.463 

12501 

0.05 

8 

2.80 

0.977 

26379 

0.11 

10 

3.18 

1.733 

46791 

0.20 

12 

3.48 

2.732 

73764 

0.31 

14 

3.79 

4.052 

109404 

0.46 

16 

4.10 

5.724 

154548 

0.65 

18 

4.33 

7.651 

206577 

0.87 

20 

4.55 

9.928 

268056 

1.13 

22 

4.79 

12.646 

341442 

1.43 

24 

5.00 

15.710 

424170 

1.78 

26 

5.20 

19.172 

517644 

2.17 

28 

5.40 

23.090 

623430 

2.62 

30 

5.63 

27.638 

746226 

3.13 

32 

5.83 

32.561 

879147 

3.69 

34 

6.00 

37.830 

1021410 

4.29 

36 

6.22 

43.969 

1187163 

4.99 

38 

6.39 

50.328 

1358856 

5.71 

40 

6.56 

57.243 

1545561 

6.49 

4? 

6.73 

64.749 

1748223 

7.34 

44 

6.87 

72.540 

1958580 

8.23 

46 

7.04 

81.249 

2193723 

9.21 

48 

7.18 

90.224 

2436048 

10.23 

WATER  SUPPLY   ENGINEERING. 


161 


TABLE  II.— (CONTINUED.) 


9 
1000 

D 

F 

Q 

O 

HP 

3 

1.68 

0.082 

2214 

0.01 

4 

1.98 

0.172 

4644 

0.02 

6 

2.50 

0.490 

13230 

0.06 

g 

2.97 

1.037 

27999 

0.12 

10 

3.37 

1.837 

49599 

0.21 

12 

3  69 

2.897 

78219 

0  33 

14 

4.02 

4.297 

116019 

0.49 

16            4.33 

6.045 

163215 

0.69 

18            4.59 

8.111 

218997 

0.92 

20 

4.83 

10  539 

284553 

1.20 

22 

5.09 

13.438 

362836 

1.52 

24 

5  30 

16.653 

449631 

1.89 

26 

5  52 

20.352 

549504 

2.31 

28 

5.73 

24.501 

661527 

2.78 

30 

5.97 

29  307 

791289 

3.32 

32 

6.18 

34.515 

931905 

3  91 

34 

6.36 

40.100 

1082700 

4.55 

36 

6.60 

46.655 

1259685 

5.29 

38 

6.78 

53.400 

1441800 

6.06 

40 

6.96 

60.733 

1639791 

6.89 

42 

7.14 

68.694 

1854738 

7.79 

44 

729 

76.975 

2078325 

8.73 

46 

7.47 

86.211 

2327697 

9  78 

48 

7.62 

95.753 

2585331 

10  86 

10 
1000  - 

I) 

V 

Q 

a 

HP 

3 

1.77 

0.087 

2349 

0.01 

4 

2  09 

0.182 

4914 

0.02 

6 

2.63 

0.515 

13905 

0.06 

8 

3.16 

1.103 

29781 

0.13 

10 

3.55 

1.935 

52245 

0  22 

12 

3.89 

3.054 

.  82458 

0.35 

14 

4.24 

4.533 

122391 

0.51 

16 

4.58 

6.394 

172638 

0.73 

18 

4.84 

8.552 

230904 

0.97 

20 

5.09 

11.106 

299862 

1.26 

22 

5.35 

14.124 

381348 

1.60 

24 

5.60 

17.595 

475065 

2.00 

26 

5.82 

21.458 

579366 

2.43 

28 

6.04 

25.823 

697221 

2.93 

30 

6.29 

30.878 

833706 

3.50 

32 

6.51 

36.358 

981666 

4.12 

34 

6.70 

42.244 

1140588 

4.79 

36 

6.96 

49.200 

1328400 

5.58 

38 

7.15 

56.313 

1520451 

6.39 

40 

7.34 

64.049 

1729323 

7.26 

42 

7.53 

72.446 

1956042 

8.22 

44 

7.68 

81.093 

2189511 

9.20 

46 

7.87 

90.828 

2452356 

10.30 

48 

8.03 

100.905 

2724435 

11.44 

162 


WATER  SUPPLY  ENGINEERING. 


TABLK  III. 


12 

RI 

u* 

R 

RI 

u* 

R 

RI 
~U* 

0.05 

0.00015800 

1.75 

0.00006863 

4.0 

0.00006715 

0.10 

0.00011200 

1.80 

0.00006856 

4.1 

0.00006712 

0.15 

0.00009667 

1.85 

0.00006849       4.2 

0.00006710 

0.20 

0.00008900 

1.90 

0.00006842 

43 

0.00006707 

0.25 

0.00008440 

1.95 

0.00006836 

4.4 

0.00006705 

0.30 

0.00008133 

2.00 

0.00006830 

4.5 

0.00006702 

0.35 

0.00007914 

2.05 

0.00006824 

4.6 

0.00006700 

0.40 

0.00007750 

2.10 

0.00006819 

4.7 

0.00006698 

0.45 

0.00007622 

2.15 

0.00006814 

4.8 

0.00006696 

0.50 

0.00007520 

2.20 

0.00006809 

4.9 

0.00006694 

0.55 

0.00007437 

2.25       0.00006804 

5.0 

0.00006692 

0.60 

0.00007367 

2.30       0.00006800 

5.1 

0.00006690 

0.65 

0.00007308 

2.35       0.00006796 

5.2 

0.00006689 

0.70 

0.00007257 

2.40       0.00006792 

5.3 

0.00006687 

0.75 

0.00007213 

2.45       0.00006788 

5.4 

0.00006685 

0.80 

0.00007175 

2.50       0.00006784 

5.5 

0.00006684 

0.85 

0.00007141 

2.55       0.00006780 

5.6 

0.00006682 

0.90 

0.00007111 

2.60       0.00006777 

5.7 

0.00006681 

0.95 

0.00007084 

2.65       0.00006774 

5.8 

0.00006679 

1.00 

0.00007060 

2.70 

0.00006770 

5.9 

0.00006678 

1.05 

0.00007038 

2.75 

0.00006767 

6.0 

0.00006677 

1.10 

0.00007018 

2.80 

0.00006764 

6.1 

0.00006675 

1.15 

0,00007000 

2.85 

0.00006761 

6.2 

0.00006874 

1.20 

O.\)0006983 

2.90 

0.00006759 

6.3 

0.00006673 

1.25 

0.00006968 

3.0 

0.00006753 

6.4 

0.00006672 

1.30 

0.00006954 

3.1 

0.00006748 

6.5 

0.00006671 

1.35 

0.00006941 

3.2 

0.00006744 

7.0 

0.00006666 

1.40 

0.00006929 

3.3 

0.00006739 

7.5 

0.00006661 

1.45 

0.00006917 

3.4 

0.00006735 

8.0 

0.00006658 

1.50 

0.00006907 

3.5 

0.00006731 

8.5 

0.00006654 

1.55 

0.00006897 

3.6 

0.00006728 

9.0 

0.00006651 

1.60 

0.00006888 

3.7 

0.00006724 

9.5 

0.00006648 

1.65 

0.00006879 

3.8 

0.00006721 

10.0 

0.00006646 

1.70 

0.00006871 

3.9 

0.00006718 

INDEX. 


Alinement  (see  pipelaying,  tunnels) 

Albuminoid  ammonia   114 

Arches: 

Abutments  for,  formula  for  calculating 143 

Best   form   of 145 

Curve  of  pressure,  calculation  of 146-149 

Design,  general  rules  governing 149-153 

Failure,   modes   of. 145 

Joint  of  rupture  of 139 

Thickness,  methods  of  calculating 137-143 

Artesian  wells,  general  discussion  of. 74,  114 

Axioms  of  hydraulics 9 

Backfilling  trenches  for  water  pipe 63 

Back-pressure,  loss  of  head  from  57 

Bends  and  elbows,  loss  of  head  by 60 

Calking: 

Directions  for    66 

Lead,  weight  required 67 

Center  walls  for  earth  dams 96 

Chlorine   114 

Coefficients,  smoothness  of  pipes 17,  18,  51 

Concrete,  composition  for  hydraulic  work 99 

Conduits,  masonry: 

Flow  through,  compared  with  pipe  lines 124 

Horse-shoe  section,  maximum  flow  of 125 

Large  sizes,  form  of  section  best  for 125 

Core  walls  for  earth  dams 96 

Croton  basin,  Rainfall  available  from 77 

Curve  of  pressure  in  arches 146-149 

Dams: 

Character  of,  suitable  to  different  conditions 82 

Concrete  masonry  work  for 99 

Foundation  beds,  determining  suitability  of 82 

Foundation  pits,  drainage  of 104 

Location,  conditions  governing 82 

Dams,  earth: 

Centre  walls  for 96 

Discharge  outlets  for 98,  122 

Embankments  for  96,  98 

Spillways  for 97 

Dams,  masonry: 

Base,  construction  of 95 

Classes  of 82 

Design  of  high, 
Example  illustrating 88-93,  122 


164  INDEX. 

Formulas  for,  unreliability  of 93 

Equilibrium  of. 

Rectangular  sections 82 

Trapezoidal  sections 83 

Hydrostatic  pressure  of  water  against 119 

Line  of  pressure,  location  of 86 

Plan,  conditions  governing  selection  of 94 

Overturning,  stability  against 82-86 

Pressure  on  foundation  masonry,  safe  amount 87,  121 

Sliding  on  base,  stability  against 117-119 

Stone  masonry  work  for , 100 

Stability,  Vauban's  principle 84 

Diameter  (see  pipes,  pipe  lines,  pipe  line  systems) 

Discharge  outlets  for  earthen  dams 98,  122 

Discharge  (see  pipes,  pipe  lines,  pipe  line  systems) 

Drainage  of  foundation  pits 104 

Duty  trials   (see  pumping  engines) 

Embankments  for  earth  dams 96,  98 

Entry,  resistance  to 

Head,  loss  of  due  to 66 

Short  horizontal  pipes 12 

Evaporation: 

From  water  surfaces . . 116 

Sources,   principal 72 

Fifth  powers,  Tables  of 48,  110 

Fifth  roots,  calculation,  use  of  logarithms  in 28 

Filters,  mechanical, 

Action,  method  of 128 

Cost  of  operation 128 

Filtration,  general  discussion  of 127 

Fire  service,  quantity  of  water  required  for 116 

Flow: 

Friction,  loss  of  head  from 56 

Horse  shoe  conduits,  maximum  through 125 

Pipe  lines,  branched, 

Branches  variously  placed,  effect  on 31-38 

Calculation,  example  illustrating , . .  28,  106 

Pipe  lines, 

Calculation,  example  illustrating 19,  21 

Fundamental  equations  for 18,  106 

Varying  diameters,  calculation  of 22 

Pipe  line  system, 

Calculation,  abbreviated  method 49,  107 

Calculation,  formulas  for 42,  106 

Communicating  with  two  reservoirs 51 

Through  open  channels 123 

Velocity  of, 

Comparative  through  masonry  conduits  and  smooth  pipes....  124 


INDEX.  165 

Grade  of  pipe  lines,  effect  on 14 

Length  of  pipe  effect  on 12 

Smoothness  of  pipe,  effect  on 17 

Short  horizontal  pipes 11 

Vertical  pipes 59 

Formulas: 

Abutments  for  arches 143 

Diameter  of  pipe  carrying  100  gallons  per  capita  in  ten  hours. . .  69 

Dimensioning   spillways 81 

Duty  of  pumping  engines 131 

Equilibrium  of  masonry  dams 82 

Flow  in  pipe  line  system 41,  42,  106 

Flow  through  long  pipes 18,  106 

Flow  through  open  channels 123 

Horse  power  for  pumping 69,  112 

Safe  pressure  on  bottom  courses  of  masonry  dams 87,  121 

Spillways,  approximate  for  determining 81 

Storage  reservoirs,  capacity  of 117 

Thickness  of  arch  keys 138 

Velocity  of  falling  bodies 11 

Weight  of  cast  iron  pipe 168 

Weight  of  lead  required  for  calking 67 

Foundation  blocks  for  water  pipe 65 

Foundations: 

Dams,  testing  suitability  of 82 

Drainage  of ; 104 

Friction  in  pipes,  head  required  to  overcome 12,  56 

Grade  (see  pipes,  pipe  lines,  pipe  line  systems,  pipelaying,  flow) 
Head: 
Definition  of 

Horizontal  pipe  lines 11 

Pipe  lines  with  varying  grades 13 

Height  of, 

Friction  in  pipes,  amount  required  to  overcome 12 

Resistance  to  entry,  amount  required  to  overcome 12 

Losses  due  to, 

Back    pressure , 57 

Bends  and  elbows 60 

Changes  in  diameter  of  pipe 59 

Friction  of  flow 56 

Resistance  to  entry 56 

Velocity,  height  required  to  produce 56 

Loss  of 

Definition  of  term 55 

Hydraulic  grade  line,  effect  on 63 

Pipe  line  systems,  calculation  of 56 

Practical  importance  of 63 


166  INDEX. 

Pressure  in  pipe  lines  due  to 14 

Heart  walls  for  earth  dams 96 

Hydraulic  grade  line: 

Correct  determination,  importance  of 14 

Definition  of  term 9 

Determination  for  pipe  lines  with  varying  grades 18 

Head,  changes  caused  by  loss  of 63 

Pipe  lines,  calculation  for 21 

Pressure  in  pipe  lines,  relation  to 25 

Varying  steepness,  effect  on  flow 14 

Hydraulic  pressure,  definition  of  term 14 

Hydraulics,  axiomatic  truths  of 9 

Hydrostatic  pressure: 

Against  plane  surfaces 119 

Definition  of  term 14 

Impurities,  mineral,  different  kinds  of 114 

Joint  of  rupture,  definition  of 139 

Line  of  pressure,  in  high  masonry  dams '. 86 

Masonry  dams  (see  dams,  masonry) 

Masonry,  stone,  rules  for  constructing 100 

Mechanical  filters  (see  filters,  mechanical) 

Mineral  impurities  in  water. 114 

Nitrates  and  nitrites 114 

Piezometric  head: 

Branched  pipe  lines,  calculation  for 28,  106 

Calculation  of,  example  illustrating 21 

Pipe  line  system,  calculation  for 49,  107 

Pipe  line  system,  communicating  with   two  reservoirs,   calcu- 
lation   for 52 

Piezometric  height,  definition  of  term 10 

Piezometric  tubes,  definition  of 14 

Pipe,  diameter  of,  having  capacity  of  100  gallons  per  capita  in  ten 

hours,  formula  for , 69 

Pipelaying: 

Alinement,  methods  of 65 

Backfilling  trench 66 

Calking,  directions  for 66 

Calking,  weight  of  lead  required 67 

Foundation  blocks  for 65 

General   discussion 65 

Grades,  method  of  preserving 65 

Pipe  lines: 
Diameter, 

Calculation,  example  illustrating 20 

Changes  in,  loss  of  head  from 59 

Flow, 

Calculation  of,  example  illustrating 19 

Fundamental  equations  for 18,  106 

Smoothness  of  pipe,  effect  on 17 


INDEX.  167 

Varying  diameter  of  pipe  effect  of 22 

Friction  in,  head  required  to  overcome 12,  56 

Head  of  (see  head) 
Hydraulic  grade  line 

Calculation,  example  illustrating 21 

Determination  for  varying  grades 13 

Length,  measurement  adapted  for  calculations 19 

Location  with  respect  to  hydraulic  grade  line 14 

Piezometric  head,  calculation  of,  example  illustrating.. 21 

Pressure  in, 

General  discussion 14 

Relation  to  hydraulic  grade  line 25 

Uniform    diameter    equivalent    to    compound    system,    calcula- 
tion of 26,  27 

Pipe  lines,  branched, 
Flow, 

Branches  variously  placed 31-38 

Calculation  of  example  illustrating 28,  106 

Pipe  line  system: 

Communicating  with  two  reservoirs 51 

Diameter  of  pipes  for,  calculation  of 39 

Flow, 

Calculation,  abbreviated  method 49,  107 

Formulas  for  calculating '. 42,  47,  106,  107 

Friction  in 12,  56 

Head,  loss  of,  calculation  of 56 

Piezometric  head,  calculation,  abbreviated  method 49,  107 

Pipes: 

Cast  iron,  tables  of  weight  and  thickness 68 

Friction  in 12,  56 

Long  horizontal,  velocity  of  flow  in 12 

Short  horizontal 

Discharge  through,  velocity  of 11 

Resistance  of  entry  to 12 

Smoothness,  coefficients  of 17 

Vertical,  flow  through 59 

Pressure  (see  pipes,  pipe  lines,  pipe  line  systems) 

Pumping  horse  power  required,  formula  for 69,  112 

Pumping  engines: 

Classes,  description  of  different 129 

Class,  selection  of,  conditions  governing 129 

Duty  of,  formulas  for 131 

Duty  trials, 

Definition  of 130 

Methods  of  conducting 130-137 

High  duty,  proper  field  of  use 130 

Types  of,  merits  of  different 137 

Pumps  (see  pumping  engines) 
Purity  (see  water  supply) 


168  INDEX. 

Rainfall: 

Amount  available  from  Croton  Basin 77 

Methods  of  estimating 77 

Per  square  mile  of  drainage  area 78 

Reservoirs,  Storage: 

Capacity  required,  conditions  governing 78 

Spillways  for,  approximate  formulas 81 

Smoothness  (see  pipes,  pipe  lines,  pipe  line  systems) 
Spillways: 

Dams,   earth 97 

Formulas,  approximate,  for 81 

Springs,  quality  of  water  supply  from 75 

Storage,  capacity  required: 

Calculation  of,  example  illustrating 78 

Formula  for  calculating 117 

Stone  masonry,  rules  regarding  construction 100 

Streams: 

Relative  purity  of  large  and  small 72 

Yield,  method  of  estimatng 77 

Tables: 

Cast  iron  pipe,  weight  and  thickness  of 68 

Coefficients  of  smoothness  of  pipes 18 

Explanation  of 154 

Fifth  powers  of  pipe  diameters 48 

Tubes,  piezometric,  definition  of 14 

Tunnels: 

Alinement,  possible  accuracy  of 126 

Construction  of 125 

Velocity  (see  flow,  head) 
Water  supply: 

Artesian  wells  for,  general  discussion 74,  114 

Evaporation,  amount  of 72,  116 

Filtration   of 127 

Impurities,  common  mineral 114 

Quality  of: 

Chemical  analysis  inadequate  to  determine 71,  113 

General  criterion  of 71 

General  discussion  on 71-76 

Large  and  small  streams 72 

Springs,  general  purity  of 75 

Wells,  impurity  of 74 

Quantity  required: 

Fire  service 116 

Future  growth  in  population,  provisions  for 76 

Per  capita  per  24  hours 76 

Sources,  classification  of 72 

Storage  for,  example  illustrating 78 

Wells: 

Artesian,  general  discussion 74,  114 

Driven,  quality  of  supply  from 74 


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